Collatz Streetmap - Revision history

imported>Gfis: export to 3x+1 Problem

2018-09-03T19:31:34Z

export to 3x+1 Problem

← Older revision		Revision as of 19:31, 3 September 2018
Line 1:		Line 1:
	==Introduction==		==Introduction==
	Collatz sequences are sequences of non-negative integer numbers with a simple construction rule: even elements a halved, and odd elements are multiplied by 3 and then incremented by 1. Since ~~many years~~ it is unknown whether the final cyle 4 - 2 - 1 is always for any start value. This problem is the '''Collatz conjecture''', for which the [https://en.wikipedia.org/wiki/Collatz_conjecture english Wikipedia] states:		Collatz sequences are sequences of non-negative integer numbers with a simple construction rule: even elements a halved, and odd elements are multiplied by 3 and then incremented by 1. Since decades it is unknown whether the final cyle 4 - 2 - 1 is always reached for any start value. This problem is the '''Collatz conjecture''', for which the [https://en.wikipedia.org/wiki/Collatz_conjecture english Wikipedia] states:
	: It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.		: It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

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	Straightforward visualizations of the Collatz graph show now obvious structure. The sequences for the first dozen of start values seem to be rather harmless, but the sequence for 27 suddenly has 112 elements.		Straightforward visualizations of the Collatz graph show now obvious structure. The sequences for the first dozen of start values seem to be rather harmless, but the sequence for 27 suddenly has 112 elements.

			Read on in [[OEIS/3x+1 Problem\|3x+1 Problem]].

	This article proposes		This article proposes

imported>Gfis: Coverage of S

2018-08-28T21:12:59Z

Coverage of S

← Older revision		Revision as of 21:12, 28 August 2018
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	+1 6+4 +1 6+4 +1		+1 6+4 +1 6+4 +1
	The construction stops since there is no number q such that q * 3 + 1 = 62.		The construction stops since there is no number q such that q * 3 + 1 = 62.
	===Street directory T===		===Street directory S===
	Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the paired elements of the streets for all possible starting values 4, 10, 16, 22 ... 6n+4 for n = 0, 1, 2 ... as rows of a table, in reversed direction (extending to the right). The elements ≡ 4 mod 6 are highlighted:		Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the paired elements of the streets for all possible starting values 4, 10, 16, 22 ... 6n-2 for n = 1, 2 ... as rows of a table, in reversed direction (extending to the right). The elements ≡ 4 mod 6 are highlighted:
	<table>		<table>
	<tr align="right">		<tr align="right">
Line 102:		Line 102:
	: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.		: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.
	====Street construction rules====		====Street construction rules====
	The following table shows the rules for the construction of the first 9 columns <nowiki>T[n,1..9]</nowiki> of row n (n = 0, 1, 2 ..) in the street directory:		The following table shows the rules for the construction of the first 9 columns <nowiki>S[n,1..9]</nowiki> of row n (n = 1, 2, 3 ...) in the street directory:
	{\| class="wikitable" style="text-align:left"		{\| class="wikitable" style="text-align:left"
	\|-		\|-
	!Column!! Steps !! Expression \|\| Formula !! Condition !! ~~Values~~		!Column!! Steps !! Expression \|\| Formula !! Condition !! Coverage
	\|-		\|-
	\| 1\|\| \|\| ~~6n+4~~ \|\| \|\| \|\| 4,10,16,22~~,28, ...~~		\| 1\|\| \|\| \|\| 6n-2 \|\| \|\| 4,10,16,22 mod 24
	\|-		\|-
	\| 2\|\| d \|\| (6n+4-1)/3 \|\| 2n+1 \|\| \|\| ~~1,3,5,7, ...~~		\| 2\|\| d \|\| (6n-2-1)/3 \|\| 2n-1 \|\| \|\| all odd numbers
	\|-		\|-
	\| 3\|\| m \|\| (6n+4)*2 \|\| 12n+8 \|\| \|\| 8,20~~,32,44, ...~~		\| 3\|\| m \|\| (6n-2)*2 \|\| 12n-4 \|\| \|\| 8,20 mod 24
	\|-		\|-
	\| 4\|\| dm \|\| ((6n~~+4)~~-1)/3*2 \|\| 4n+2 \|\| \|\| 2,6,10,14, ~~...~~		\| 4\|\| dm \|\| ((6n-2-1)/3)*2 \|\| 4n-2 \|\| \|\| 2,6,10,14,18,22 mod 24
	\|-		\|-
	\| 5\|\| mm \|\| (6n+4)22 \|\| 24n~~+16~~ \|\| \|\| 16~~,40,64,88, ...~~		\| 5\|\| mm \|\| (6n-2)22 \|\| 24n-8 \|\| \|\| 16 mod 24
	\|-		\|-
	\| 6\|\| dmm \|\| (((6n~~+4)~~-1)/3)22 \|\| 8n+4 \|\| \|\| 4,12,20~~,28, ...~~		\| 6\|\| dmm \|\| ((6n-2-1)/3)22 \|\| 8n-4 \|\| \|\| 4,12,20 mod 24
	\|-		\|-
	\| 7\|\| mmd \|\| ((6n+4)*4-1)/3 \|\| 8n+5 \|\| \|\| 5,13,21~~,29, ...~~		\| 7\|\| mmd \|\| ((6n-2)22-1)/3 \|\| 8n-3 \|\| \|\| 5,13,21 mod 24
	\|-		\|-
	\| 8\|\| dmmd \|\| (((6n~~+4)~~-1)/3)22-1)/3 \|\| 8n/3+1 \|\| n ≡ 0 mod 3 \|\| 1,9,17,25 ...		\| 8\|\| dmmd \|\| ((6n-2-1)/3)22-1)/3 \|\| (8n-5)/3 \|\| n ≡ 1 mod 3 \|\| (1,9,17,25 ...)
	\|-		\|-
	\| 9\|\| mmdm \|\| (((6n+4)22-1)/3)*2 \|\| 16n~~+10~~ \|\| n ≡ 0 mod 3 \|\| 10,58,106,154, ...		\| 9\|\| mmdm \|\| ((6n-2)22-1)/3)*2 \|\| 16n-6 \|\| n ≡ 1 mod 3 \|\| (10,58,106,154, ...)
	\|}		\|}
	~~====Street lengths > 7====~~		The first 6 columns of the table cover the odd numbers and all numbers ≡ 2,4,6,8,10,12,16,18,20,22 mod 24.
	~~For indexes n~~ &#~~x2262~~; ~~0 mod 3~~, ~~the construction stops at column 7. We therefore concentrate on the indexes divisible by 3 and make the following observations:~~
	* The lengths of all streets are finite.
	** <nowiki>T[n,~~i] dm T[n~~,~~i+4] for i=5~~,9,13,~~17 .~~.~~.</nowiki> and~~

	----		It is ''not shown'' so far that all multiples of 24 are contained in the table.
			<!-- ???
			All odd multiples of 3 are contained in column 2. All multiples of 24 can be reached by duplicating them 3 times (3 m-steps).
	The highlighted ~~pairs of numbers~~ ≡ 4 mod 6 are ~~decreasing to~~ the ~~right~~.		-->
	* The lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the street construction rule.		====Highlighted numbers====
			The numbers of the form 6p-2 were highlighted in the example above. They have the special property that, when p > 0 and p ≡ 0 mod 3, a dm-step yields a number of the same form, but with one factor 3 in p replaced by 2:
			6(3q)-2 dm ((6(3q)-2)-1)/32 = (6q-1)2 = 6(2q)-2
			This implies that a dm-step decreases any number by about one third.
			====Street lengths > 7====
			* Columns 4(k+1)+1 result by dm-steps from columns 4k+1 for k=1,2,... (and so do columns 4(k+1)+2 result from columns 4k+2). Sequences of dm-steps decrease the numbers. Therefore the lengths of all streets are finite.
			* Column 5 is 24n-8, and the lengths depend on the power of 3 contained in that n.
			<!--
			* The street lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the street construction rule.
	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"
	\| 4 mod 54		\| 4 mod 54
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	\| 3		\| 3
	\|}		\|}
	* At the starting values 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]) the street lengths ~~n<sub>i</sub>~~ have high values 5, 9, 13, 17, 21 which did not occur before. Those starting values are (9<sup>n+1</sup> - 1) / 2, or 4 * Sum(9<sup>i</sup>, i=0..n).		-->
	* The pattern of increasing and decreasing lengths is replicated when subsets of the rows (mod 9, 27, 81 ...) are regarded.		* At the starting values 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]) the street lengths have high values 5, 9, 13, 17, 21 which did not occur before. Those starting values are (9<sup>n+1</sup> - 1) / 2, or 4 * Sum(9<sup>i</sup>, i=0..n).
	====Coverage====		====Coverage====
	The elements of the streets are strongly interconnected, and the table "obviously" shows all positive integers which are not multiples of 24:		The elements of the streets are strongly interconnected, and the table "obviously" shows all positive integers which are not multiples of 24:
	{\| class="wikitable"		{\| class="wikitable"
	\| r<sub>0</sub> ≡ 4 mod 6		\| r<sub>1</sub> ≡ 4 mod 6
	\| style="text-align:right" \| ≡ 4,10,16,22 mod 24		\| style="text-align:right" \| ≡ 4,10,16,22 mod 24
			\|-
			\| r<sub>2</sub> ≡ 1 mod 2
			\| all odd numbers
	\|-		\|-
	\| r<sub>3</sub> ≡ 8 mod 12		\| r<sub>3</sub> ≡ 8 mod 12
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	\| r<sub>6</sub> ≡ 4 mod 8		\| r<sub>6</sub> ≡ 4 mod 8
	\| style="text-align:right" \| ≡ 4,12,20 mod 24		\| style="text-align:right" \| ≡ 4,12,20 mod 24
	\|-
	~~\| r<sub>2</sub> ≡ 1 mod 2~~
	~~\| all odd numbers~~
	\|}		\|}
	~~All odd multiples of 3 are contained in column r<sub>2</sub>. All multiples of 24 can be reached by duplicating them 3 times (3 m-steps).~~

	So if we can show that we reach all start values ≡ 4 mod 6, we have a proof that all positive integers are reached.		So if we can show that we reach all start values ≡ 4 mod 6, we have a proof that all positive integers are reached.

imported>Gfis: street lengths > 7

2018-08-28T13:07:11Z

street lengths > 7

← Older revision		Revision as of 13:07, 28 August 2018
Line 1:		Line 1:
	==Introduction==		==Introduction==
	Collatz sequences are sequences of non-negative integer numbers with a simple construction rule: even elements a halved, and odd elements are multiplied by 3 and then incremented by 1. Since many years it is unknown whether the final cyle 4 - 2 - 1 is always for any start value. This problem is the '''Collatz conjecture'''., for which the [https://en.wikipedia.org/wiki/Collatz_conjecture english Wikipedia] states: It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.		Collatz sequences are sequences of non-negative integer numbers with a simple construction rule: even elements a halved, and odd elements are multiplied by 3 and then incremented by 1. Since many years it is unknown whether the final cyle 4 - 2 - 1 is always for any start value. This problem is the '''Collatz conjecture''', for which the [https://en.wikipedia.org/wiki/Collatz_conjecture english Wikipedia] states:
			: It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

	~~Whenever~~ we speak of ''numbers'' in this article, we mean natural~~, positive~~ integer numbers ~~(without~~ 0).		When we speak of ''numbers'' in this article, we normally mean natural integer numbers > 0. The zero is sometimes mentioned explicetely.
	===References===		===References===
	* Jeffry C. Lagarias, Ed.: ''The Ultimate Challenge: The 3x+1 Problem'', Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. [http://www.ams.org/bookpages/mbk-78 MBK78]		* Jeffry C. Lagarias, Ed.: ''The Ultimate Challenge: The 3x+1 Problem'', Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. [http://www.ams.org/bookpages/mbk-78 MBK78]
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	+1 6+4 +1 6+4 +1		+1 6+4 +1 6+4 +1
	The construction stops since there is no number q such that q * 3 + 1 = 62.		The construction stops since there is no number q such that q * 3 + 1 = 62.
	===Street directory===		===Street directory T===
	Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the paired elements of the streets for all possible starting values 4, 10, 16, 22 ... ~~n*6~~+4 as rows of a table, in reversed direction (extending to the right). The ~~columns of the rows are named c<sub>1</sub>, c<sub>2</sub>, c<sub>3</sub> ..., and the~~ elements ≡ 4 mod 6 are highlighted:		Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the paired elements of the streets for all possible starting values 4, 10, 16, 22 ... 6n+4 for n = 0, 1, 2 ... as rows of a table, in reversed direction (extending to the right). The elements ≡ 4 mod 6 are highlighted:
	<table>		<table>
	<tr align="right">		<tr align="right">
Line 101:		Line 102:
	: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.		: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.
	====Street construction rules====		====Street construction rules====
	The following table shows the rules for the construction of the columns in the street directory:		The following table shows the rules for the construction of the first 9 columns <nowiki>T[n,1..9]</nowiki> of row n (n = 0, 1, 2 ..) in the street directory:
	{\| class="wikitable" style="text-align:left"		{\| class="wikitable" style="text-align:left"
	!Column !! Steps !! Formula !! ~~Increment~~ !! Values		\|-
	\|-		!Column!! Steps !! Expression \|\| Formula !! Condition !! Values
	\| ~~c<sub>~~1~~</sub>~~ \|\| \|\| ~~c<sub>1</sub> =~~ 4 ~~+ 6*n~~ \|\| 6 \|\| 4,10,16,22,28, ...		\|-
	\|-		\| 1\|\| \|\| 6n+4 \|\| \|\| \|\| 4,10,16,22,28, ...
	\| ~~c<sub>~~2~~</sub>~~ \|\| d \|\| ~~c<sub>2</sub> =~~ (~~c<sub>1</sub>~~ - 1) / 3 \|\| 2 \|\| 1,3,5,7, ...		\|-
	\|-		\| 2\|\| d \|\| (6n+4-1)/3 \|\| 2n+1 \|\| \|\| 1,3,5,7, ...
	\| ~~c<sub>~~3~~</sub>~~ \|\| m \|\| ~~c<sub>3</sub> = c<sub>1</sub>~~ * 2 \|\| 12 \|\| 8,20,32,44, ...		\|-
	\|-		\| 3\|\| m \|\| (6n+4)*2 \|\| 12n+8 \|\| \|\| 8,20,32,44, ...
	\| ~~c<sub>~~4~~</sub>~~ \|\| dm \|\| ~~c<sub>~~4~~</sub> = (c<sub>1</sub>~~ - 1) / 3 * 2 \|\| 12 \|\| 2,6,10,14, ...		\|-
	\|-		\| 4\|\| dm \|\| ((6n+4)-1)/3*2 \|\| 4n+2 \|\| \|\| 2,6,10,14, ...
	\| ~~c<sub>~~5~~</sub>~~ \|\| mm \|\| ~~c<sub>5</sub> = c<sub>1</sub>~~ * 4 \|\| 24 \|\| 16,40,64,88, ...		\|-
	\|-		\| 5\|\| mm \|\| (6n+4)22 \|\| 24n+16 \|\| \|\| 16,40,64,88, ...
	\| ~~c<sub>~~6~~</sub>~~ \|\| dmm \|\| ~~c<sub>6</sub> =~~ (~~c<sub>1</sub>~~ - 1) / 3 * 4 \|\| 8 \|\| 4,12,20,28, ...		\|-
	\|-		\| 6\|\| dmm \|\| (((6n+4)-1)/3)22 \|\| 8n+4 \|\| \|\| 4,12,20,28, ...
	\| ~~c<sub>~~7~~</sub>~~ \|\| mmd \|\| ~~c<sub>7</sub> =~~ (~~c<sub>1</sub>~~ * 4 - 1) / 3 \|\| 8 \|\| 5,13,21,29, ...		\|-
	\|-		\| 7\|\| mmd \|\| ((6n+4)*4-1)/3 \|\| 8n+5 \|\| \|\| 5,13,21,29, ...
	\| ~~c<sub>~~8~~</sub>~~ \|\| dmmd \|\| ~~c<sub>6</sub> =~~ ((~~c<sub>1</sub>~~ - 1) / 3 * 4 - 1) / 3 \|\| 16 \|\| 4,12,~~20,28~~, ...		\|-
	\|-		\| 8\|\| dmmd \|\| (((6n+4)-1)/3)22-1)/3 \|\| 8n/3+1 \|\| n ≡ 0 mod 3 \|\| 1,9,17,25 ...
	\| ~~c<sub>~~9~~</sub>~~ \|\| mmdm \|\| ~~c<sub>7</sub> =~~ (~~c<sub>1</sub>~~ * 4 - 1) / 3 * 2 \|\| 8 \|\| 5,13,21,29, ...		\|-
			\| 9\|\| mmdm \|\| (((6n+4)22-1)/3)*2 \|\| 16n+10 \|\| n ≡ 0 mod 3 \|\| 10,58,106,154, ...
	\|}		\|}
	====Street lengths====		====Street lengths > 7====
	* The lengths of ~~the~~ streets ~~seem to be~~ finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the ~~end of the street~~.		For indexes n ≢ 0 mod 3, the construction stops at column 7. We therefore concentrate on the indexes divisible by 3 and make the following observations:
			* The lengths of all streets are finite.
			** <nowiki>T[n,i] dm T[n,i+4] for i=5,9,13,17 ...</nowiki> and

			----


			The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the right.
	* The lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the street construction rule.		* The lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the street construction rule.
	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"

imported>Gfis: Gfis moved page Collatz sequences to Collatz Streetmap without leaving a redirect

2018-08-28T11:32:27Z

Gfis moved page Collatz sequences to Collatz Streetmap without leaving a redirect

← Older revision		Revision as of 11:32, 28 August 2018
(No difference)

imported>Gfis: street rules table

2018-08-28T10:23:11Z

street rules table

← Older revision		Revision as of 10:23, 28 August 2018
Line 43:		Line 43:
	A starting number and a sequence of step names defines a unique, directed path in the Collatz graph.		A starting number and a sequence of step names defines a unique, directed path in the Collatz graph.
	=== Trivial paths===		=== Trivial paths===
	There are two ~~cases~~ of paths ~~with a~~ very simple ~~structure~~:		There are two types of paths whose descriptions are very simple:
	* Powers of 2: (n = 2<sup>k) hhhh ... h 8 h 4 h 2 h 1		(n = 2<sup>k</sup>) hhhh ... h 8 h 4 h 2 h 1 - powers of 2
	* Numbers divisible by 3~~: n~~ mmm ... m (n * 2<sup>k) ...		(n ≡ 0 mod 3) mmm ... m (n * 2<sup>k</sup>) ... - multiples of 3
	==Collatz streets==		==Collatz streets==
	===Motivation: Patterns in sequences with same length===		===Motivation: Patterns in sequences with same length===
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	+1 6+4 +1 6+4 +1 6+4 +1 6+4 *6+2 +0 +0 ...		+1 6+4 +1 6+4 +1 6+4 +1 6+4 *6+2 +0 +0 ...
	The third line tells how the second line can be computed from the first.		The third line tells how the second line can be computed from the first.
			Walking from right to left, the step pattern is:
	Informally, a '''street''' is a parallel arrangement of 2 paths stemming from 2 sequences which have a common tail. A street starts with an element ≡ 4 mod 6 (364 in the example, before the bar), it proceeds to the left with a d/m and a m/m ~~step~~, and then it extends to the left as long as a characteristical, alternating sequence of pairs of steps m/d - d/m - m/d - d/m ... can be continued. In the example, the street can be continued with 4 additional steps.		d m m d m d m d m
			m m d m d m d m d
			or, in linearized form:
			d/m m/m m/d d/m m/d d/m m/d d/m m/d ...
			Informally, a '''street''' is a parallel arrangement of 2 paths stemming from 2 sequences which have a common tail. A street starts with an element ≡ 4 mod 6 (364 in the example, before the bar), it proceeds to the left with a d/m and a m/m pair of steps, and then it extends to the left as long as a characteristical, alternating sequence of pairs of steps m/d - d/m - m/d - d/m ... can be continued. In the example, the street can be continued with 4 additional pairs of steps:
	q t \| 62 h 31 t 94 h 47 t 142 h ...		q t \| 62 h 31 t 94 h 47 t 142 h ...
	126 h \| 63 t 190 h 95 t 286 h 143 t ...		126 h \| 63 t 190 h 95 t 286 h 143 t ...
	+1 6+4 +1 6+4 +1		+1 6+4 +1 6+4 +1
	The construction stops since there is no number q such that q * 3 + 1 = 62:		The construction stops since there is no number q such that q * 3 + 1 = 62.
	===Street directory===		===Street directory===
	Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the streets for all possible starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are named c<sub>1</sub>, c<sub>2</sub>, c<sub>3</sub> ..., and the elements ≡ 4 mod 6 are highlighted.		Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the paired elements of the streets for all possible starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are named c<sub>1</sub>, c<sub>2</sub>, c<sub>3</sub> ..., and the elements ≡ 4 mod 6 are highlighted:
	<table>		<table>
	<tr align="right">		<tr align="right">
Line 94:		Line 98:
	<tr align="right"><td><strong>76</strong></td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>		<tr align="right"><td><strong>76</strong></td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>
	</table>		</table>

	There is a more '''[http://www.teherba.org/fasces/oeis/collatz/roads.html elaborated example]''' for elements <= 143248.		There is a more '''[http://www.teherba.org/fasces/oeis/collatz/roads.html elaborated example]''' for elements <= 143248.
	: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.		: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.
			====Street construction rules====
			The following table shows the rules for the construction of the columns in the street directory:
			{\| class="wikitable" style="text-align:left"
			!Column !! Steps !! Formula !! Increment !! Values
			\|-
			\| c<sub>1</sub> \|\| \|\| c<sub>1</sub> = 4 + 6*n \|\| 6 \|\| 4,10,16,22,28, ...
			\|-
			\| c<sub>2</sub> \|\| d \|\| c<sub>2</sub> = (c<sub>1</sub> - 1) / 3 \|\| 2 \|\| 1,3,5,7, ...
			\|-
			\| c<sub>3</sub> \|\| m \|\| c<sub>3</sub> = c<sub>1</sub> * 2 \|\| 12 \|\| 8,20,32,44, ...
			\|-
			\| c<sub>4</sub> \|\| dm \|\| c<sub>4</sub> = (c<sub>1</sub> - 1) / 3 * 2 \|\| 12 \|\| 2,6,10,14, ...
			\|-
			\| c<sub>5</sub> \|\| mm \|\| c<sub>5</sub> = c<sub>1</sub> * 4 \|\| 24 \|\| 16,40,64,88, ...
			\|-
			\| c<sub>6</sub> \|\| dmm \|\| c<sub>6</sub> = (c<sub>1</sub> - 1) / 3 * 4 \|\| 8 \|\| 4,12,20,28, ...
			\|-
			\| c<sub>7</sub> \|\| mmd \|\| c<sub>7</sub> = (c<sub>1</sub> * 4 - 1) / 3 \|\| 8 \|\| 5,13,21,29, ...
			\|-
			\| c<sub>8</sub> \|\| dmmd \|\| c<sub>6</sub> = ((c<sub>1</sub> - 1) / 3 * 4 - 1) / 3 \|\| 16 \|\| 4,12,20,28, ...
			\|-
			\| c<sub>9</sub> \|\| mmdm \|\| c<sub>7</sub> = (c<sub>1</sub> * 4 - 1) / 3 * 2 \|\| 8 \|\| 5,13,21,29, ...
			\|}
	====Street lengths====		====Street lengths====
	* The lengths of the streets seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the street.		* The lengths of the streets seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the street.

imported>Gfis: directory without lengths

2018-08-28T08:09:17Z

directory without lengths

imported>Gfis: Streets

2018-08-28T08:03:45Z

Streets

Show changes

imported>Gfis: new def. of road

2018-08-27T21:28:50Z

new def. of road

← Older revision		Revision as of 21:28, 27 August 2018
Line 25:		Line 25:
	* such that the reader could finally be convinced that the algorithm enumerates all numbers.		* such that the reader could finally be convinced that the algorithm enumerates all numbers.
	===Steps===		===Steps===
			In order to walk back and forth in the Collatz graph, we need to describe how to move from one node in the graph to another. We will write
			a ''s'' b
			when use step ''s'' to move from node (element) a in the Collatz graph/sequence to node b. The following table lists the steps which we will use:
			{\| class="wikitable" style="text-align:center"
			!Name !! Mnemonic !! Direction !! Operation !! Condition !! Remark
			\|-
			\| h \|\| halve \|\| -> root \|\| b = a / 2 \|\| a ≡ 0 mod 2 \|\| next in sequence
			\|-
			\| t \|\| triple \|\| -> root \|\| b = 3 * a + 1 \|\| true \|\| next in sequence
			\|-
			\| m \|\| multiply \|\| root -> \|\| b = a * 2 \|\| true \|\| next up in tree
			\|-
			\| d \|\| divide \|\| root -> \|\| b = (a - 1) / 3 \|\| a ≡ 1 mod 3 \|\| next up in tree
			\|}
			==Collatz roads==
	===Motivation: Patterns in sequences with same length===		===Motivation: Patterns in sequences with same length===
	When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length~~, and with a characteristical neighbourhood of every other value~~, for example (from [https://oeis.org/A070165 OEIS A070165]):		When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, for example (from [https://oeis.org/A070165 OEIS A070165]):
	142/104: [142 m 71 d 214 m 107 d 322 m 161 d 484 m 242 m 121 d \| 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]		142/104: [142 h 71 t 214 h 107 t 322 h 161 t 484 h 242 h 121 t 364 \| 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
			143/104: [143 t 430 h 215 t 646 h 323 t 970 h 485 t 1456 h 728 h 364 \| 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
	+1 6+4 +1 6+4 +1 6+4 +1 6+4 *6+2 = = ...		+1 6+4 +1 6+4 +1 6+4 +1 6+4 *6+2 = = ...
	~~143/104: [143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m \| 364, 182, 91, ..~~. ~~10, 5, 16, 8, 4, 2, 1]~~		The third line shows the relation between the values in the first two lines.
	~~==Collatz roads==~~
	~~We define~~ a "road~~" (with 2~~ parallel ~~"lanes") as a sequence~~ of ~~pairs of elements (in~~ 2 ~~Collatz~~ sequences ~~with adjacent start values, read from right to left)~~. ~~A road is constructed by taking some n (364 in~~ the example, the ~~last common~~ element ~~of the 2 sequences) with n~~ ≡ 4 mod 6, and ~~by applying~~ the steps		Informally, a '''road''' is a parallel arrangement of 2 pathes stemming from 2 sequences which have a common tail. In the example above, the road starts with an element ≡ 4 mod 6 (the 364 before the bar), followed by t/h and h/h steps, and the road then extends to the left as long as a characteristical, alternating sequence of pairs of steps ht - th - ht - th ... can be continued. In the example, the road can be continued to the left with 4 additional steps, but the construction then stops since there is no number q such that q * 3 + 1 = 62:
	~~d m m d m d m d~~ ...		q t \| 62 h 31 t 94 h 47 t 142 h ...
	~~m m d m d m d m~~ ...		126 h \| 63 t 190 h 95 t 286 h 143 t ...
	~~in alternating sequence~~, ~~until one of~~ the ~~elements in~~ the ~~pairs becomes divisible by 3. The construction yields one road~~ with ~~an upper lane (left elements of~~ the ~~pairs) and a lower lane (right elements), for example~~:		+1 6+4 +1 6+4 +1
	~~''364'': (121,728), (242,''1456''), (''484'',485), (161,''970''), (''322'',323), (107,''646''), (''214'',215), (71,''430''), (''~~142~~'',143)~~ ...
	~~The elements which are ≡ 4 mod 6 are emphasized. The construction continues until~~ 63 ~~≡ 0 mod 3:~~
	... ~~(47,''286''), (''94'',95), (31,''190''), (62,63).~~
	===Roads table R===		===Roads table R===
	For ~~better~~ handling (e.g. in Excel), the roads for all starting values 4, 10, 16, 22 ... n*6+4 ~~are listed~~ as rows of a table. The columns in the rows are numbered r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub> ...		For easier handling (e.g. in Excel) we list the roads for all starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are numbered r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub> ...
	<table>		<table>
	<tr align="right">		<tr align="right">
Line 113:		Line 125:
	<tr align="right"><td><strong>76</strong></td><td class="arr">4</td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>		<tr align="right"><td><strong>76</strong></td><td class="arr">4</td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>
	</table>		</table>
	~~There is a more '''[http://www.teherba.org/fasces/oeis/collatz/~~roads.~~html elaborated example]''' for elements <= 143248. The zoom factor of~~ the ~~web browser may be reduced (with Ctrl~~ "-", ~~to 25 % for example) such that the structure of~~ the ~~lengths of~~ roads ~~can be seen.~~		The roads contain 2 parallel, short, interrelated paths in the Collatz graph. Though the graph usually has a "chaotic" appearance, the roads exhibit quite some amount of regular structure.
	~~===Relation to the Collatz graph===~~
	~~The Collatz graph is the union~~ of ~~all possible Collatz sequences, read backwards. Apart from the initial cycle 1-2-4-1 it should be a tree, and it should contain all positive numbers~~.

	~~Each road contains 2 short paths~~ in ~~the Collatz graph~~, the ~~root is left of the starting value of the road, and both proceed~~ "~~upwards~~" ~~in the tree. In contrast~~ to the ~~usual "chaotic" appearance~~ of the ~~Collatz graph, there is an amount~~ of ~~obvious structure in the road construction~~.		There is a more '''[http://www.teherba.org/fasces/oeis/collatz/roads.html elaborated example]''' for elements <= 143248.
			: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of roads can be seen.
	====Road lengths====		====Road lengths====
	* The lengths r<sub>1</sub> of the roads seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the road.		* The lengths r<sub>1</sub> of the roads seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the road.

imported>Gfis: Intro rewritten

2018-08-27T20:01:51Z

Intro rewritten

@@ Line 1: / Line 1: @@
 ==Introduction==
-When all Collatz sequences are read backwards, they form a graph starting with 1, 2 ..., hopefully without cycles (except for 1,2,4,1,2,4 ...). At each node n in the graph, the path starting at the root (4) and  with the last node n can in principle be continued to 2 new nodes by a
+Collatz sequences are sequences of non-negative integer numbers with a simple construction rule: even elements a halved, and odd elements are multiplied by 3 and then incremented by 1. Since many years it is unknown whether the final cyle 4 - 2 - 1 is always for any start value. This problem is the '''Collatz conjecture'''., for which the [https://en.wikipedia.org/wiki/Collatz_conjecture english Wikipedia] states: It is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.
-* "m"-step: n * 2 (which is always possible), or a
-* "d"-step: (n - 1) / 3 (which is possible only if n &#x2261; 1 mod 3).
+Whenever we speak of ''numbers'' in this article, we mean natural, positive integer numbers (without 0).
 ===References===
 * Jeffry C. Lagarias, Ed.: ''The Ultimate Challenge: The 3x+1 Problem'', Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. [http://www.ams.org/bookpages/mbk-78 MBK78]
@@ Line 9: / Line 8: @@
 * Gottfried Helms: ''[http://go.helms-net.de/math/collatz/aboutloop/collatzgraphs.htm The Collatz-Problem]''. A view into some 3x+1-trees and a new fractal graphic representation. Univ. Kassel.
 * Klaus Brennecke: ''[https://de.wikibooks.org/wiki/Collatzfolgen_und_Schachbrett Collatzfolgen und Schachbrett]'', on Wikibooks
 ===Motivation: Patterns in sequences with same length===
 When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, and with a characteristical neighbourhood of every other value, for example (from [https://oeis.org/A070165 OEIS A070165]):

imported>Gfis: S complete

2018-08-26T19:47:20Z

S complete

← Older revision		Revision as of 19:47, 26 August 2018
Line 226:		Line 226:
	\| style="text-align:right" \| 16, 80, ...		\| style="text-align:right" \| 16, 80, ...
	\|}		\|}
	This shows that the columns s<sub>4</sub> ... s<sub>7</sub> contain all numbers ≡ 2,4,6,10,12,14 mod 16, but ~~that~~ &#x2261 0,8 mod 16 are missing so far. The ones &#x2261 8 mod 16 show up in s<sub>8</sub> resp. s<sub>9</sub>, ~~but~~ multiples of 16 ~~would be~~ missing ~~then~~, etc.		This shows that the columns s<sub>4</sub> ... s<sub>7</sub> contain all numbers ≡ 2,4,6,10,12,14 mod 16, but those ≡ 0,8 mod 16 are missing so far. The ones ≡ 8 mod 16 show up in s<sub>8</sub> resp. s<sub>9</sub>, half of the multiples of 16 are in s<sub>10</sub> resp. s<sub>11</sub> but ≡ 0,32 mod 64 are missing, etc.

			Since s<sub>2</sub> contains arbitray high powers of 3, S has rows of arbitrary length, and for the missing multiples of powers of 2 the exponents can be driven above all limits.
			: Thus S contains all positive integers.
	===Can S be generated starting at 1?===		===Can S be generated starting at 1?===
	We ask for an iterative process which starts with the row of S for index 1:		We ask for an iterative process which starts with the row of S for index 1:

← Older revision		Revision as of 08:09, 28 August 2018
Line 60:		Line 60:
	The construction stops since there is no number q such that q * 3 + 1 = 62:		The construction stops since there is no number q such that q * 3 + 1 = 62:
	===Street directory===		===Street directory===
	Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the streets for all possible starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are named c<sub>1</sub>, c<sub>2</sub>, c<sub>3</sub> ...		Though the graph usually has a "chaotic" appearance, the streets exhibit quite some amount of regular structure. This can be seen if we list the streets for all possible starting values 4, 10, 16, 22 ... n*6+4 as rows of a table, in reversed direction (extending to the right). The columns of the rows are named c<sub>1</sub>, c<sub>2</sub>, c<sub>3</sub> ..., and the elements ≡ 4 mod 6 are highlighted.
	<table>		<table>
	<tr align="right">		<tr align="right">
Line 80:		Line 80:
	<td class="arl">...</td>		<td class="arl">...</td>
	</tr>		</tr>
	<tr align="right"><td><strong>4</strong>~~</td><td class="arr">5~~</td><td class="d1">1</td><td class="d2">8</td><td class="d2">2</td><td><strong>16</strong></td><td><strong>4</strong></td><td class="d5">5</td><td class="d1">1</td><td><strong>10</strong></td><td class="d2">2</td><td class="d3">3</td></tr>		<tr align="right"><td><strong>4 </strong></td><td class="d1">1</td><td class="d2">8</td><td class="d2">2</td><td><strong>16</strong></td><td><strong>4</strong></td><td class="d5">5</td><td class="d1">1</td><td><strong>10</strong></td><td class="d2">2</td><td class="d3">3</td></tr>
	<tr align="right"><td><strong>10</strong>~~</td><td class="arr">3~~</td><td class="d3">3</td><td class="d2">20</td><td class="d0">6</td><td><strong>40</strong></td><td class="d0">12</td><td class="d1">13</td></tr>		<tr align="right"><td><strong>10</strong></td><td class="d3">3</td><td class="d2">20</td><td class="d0">6</td><td><strong>40</strong></td><td class="d0">12</td><td class="d1">13</td></tr>
	<tr align="right"><td><strong>16</strong></td><td class="~~arr~~">3</td><td class="~~d5">5</td><td class="d2~~">32</td><td><strong>10</strong></td><td><strong>64</strong></td><td class="d2">20</td><td class="d3">21</td></tr>		<tr align="right"><td><strong>16</strong></td><td class="d5">5</td><td class="d2">32</td><td><strong>10</strong></td><td><strong>64</strong></td><td class="d2">20</td><td class="d3">21</td></tr>
	<tr align="right"><td><strong>22</strong>~~</td><td class="arr">4~~</td><td class="d1">7</td><td class="d2">44</td><td class="d2">14</td><td><strong>88</strong></td><td><strong>28</strong></td><td class="d5">29</td><td class="d3">9</td><td><strong>58</strong></td></tr>		<tr align="right"><td><strong>22</strong></td><td class="d1">7</td><td class="d2">44</td><td class="d2">14</td><td><strong>88</strong></td><td><strong>28</strong></td><td class="d5">29</td><td class="d3">9</td><td><strong>58</strong></td></tr>
	<tr align="right"><td><strong>28</strong></td><td class="~~arr~~">~~3</td><td class="d3">9~~</td><td class="d2">56</td><td class="d0">18</td><td><strong>112</strong></td><td class="d0">36</td><td class="d1">37</td></tr>		<tr align="right"><td><strong>28</strong></td><td class="d3">9</td><td class="d2">56</td><td class="d0">18</td><td><strong>112</strong></td><td class="d0">36</td><td class="d1">37</td></tr>
	<tr align="right"><td><strong>34</strong>~~</td><td class="arr">3~~</td><td class="d5">11</td><td class="d2">68</td><td><strong>22</strong></td><td><strong>136</strong></td><td class="d2">44</td><td class="d3">45</td></tr>		<tr align="right"><td><strong>34</strong></td><td class="d5">11</td><td class="d2">68</td><td><strong>22</strong></td><td><strong>136</strong></td><td class="d2">44</td><td class="d3">45</td></tr>
	<tr align="right"><td><strong>40</strong>~~</td><td class="arr">9~~</td><td class="d1">13</td><td class="d2">80</td><td class="d2">26</td><td><strong>160</strong></td><td><strong>52</strong></td><td class="d5">53</td><td class="d5">17</td><td><strong>106</strong></td><td><strong>34</strong></td><td class="d5">35</td><td class="d5">11</td><td><strong>70</strong></td><td><strong>22</strong></td><td class="d5">23</td><td class="d1">7</td><td><strong>46</strong></td><td class="d2">14</td><td class="d3">15</td></tr>		<tr align="right"><td><strong>40</strong></td><td class="d1">13</td><td class="d2">80</td><td class="d2">26</td><td><strong>160</strong></td><td><strong>52</strong></td><td class="d5">53</td><td class="d5">17</td><td><strong>106</strong></td><td><strong>34</strong></td><td class="d5">35</td><td class="d5">11</td><td><strong>70</strong></td><td><strong>22</strong></td><td class="d5">23</td><td class="d1">7</td><td><strong>46</strong></td><td class="d2">14</td><td class="d3">15</td></tr>
	<tr align="right"><td><strong>46</strong>~~</td><td class="arr">3~~</td><td class="d3">15</td><td class="d2">92</td><td class="d0">30</td><td><strong>184</strong></td><td class="d0">60</td><td class="d1">61</td></tr>		<tr align="right"><td><strong>46</strong></td><td class="d3">15</td><td class="d2">92</td><td class="d0">30</td><td><strong>184</strong></td><td class="d0">60</td><td class="d1">61</td></tr>
	<tr align="right"><td><strong>52</strong>~~</td><td class="arr">3~~</td><td class="d5">17</td><td class="d2">104</td><td><strong>34</strong></td><td><strong>208</strong></td><td class="d2">68</td><td class="d3">69</td></tr>		<tr align="right"><td><strong>52</strong></td><td class="d5">17</td><td class="d2">104</td><td><strong>34</strong></td><td><strong>208</strong></td><td class="d2">68</td><td class="d3">69</td></tr>
	<tr align="right"><td><strong>58</strong>~~</td><td class="arr">5~~</td><td class="d1">19</td><td class="d2">116</td><td class="d2">38</td><td><strong>232</strong></td><td><strong>76</strong></td><td class="d5">77</td><td class="d1">25</td><td><strong>154</strong></td><td class="d2">50</td><td class="d3">51</td></tr>		<tr align="right"><td><strong>58</strong></td><td class="d1">19</td><td class="d2">116</td><td class="d2">38</td><td><strong>232</strong></td><td><strong>76</strong></td><td class="d5">77</td><td class="d1">25</td><td><strong>154</strong></td><td class="d2">50</td><td class="d3">51</td></tr>
	<tr align="right"><td><strong>64</strong>~~</td><td class="arr">3~~</td><td class="d3">21</td><td class="d2">128</td><td class="d0">42</td><td><strong>256</strong></td><td class="d0">84</td><td class="d1">85</td></tr>		<tr align="right"><td><strong>64</strong></td><td class="d3">21</td><td class="d2">128</td><td class="d0">42</td><td><strong>256</strong></td><td class="d0">84</td><td class="d1">85</td></tr>
	<tr align="right"><td><strong>70</strong>~~</td><td class="arr">3~~</td><td class="d5">23</td><td class="d2">140</td><td><strong>46</strong></td><td><strong>280</strong></td><td class="d2">92</td><td class="d3">93</td></tr>		<tr align="right"><td><strong>70</strong></td><td class="d5">23</td><td class="d2">140</td><td><strong>46</strong></td><td><strong>280</strong></td><td class="d2">92</td><td class="d3">93</td></tr>
	<tr align="right"><td><strong>76</strong>~~</td><td class="arr">4~~</td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>		<tr align="right"><td><strong>76</strong></td><td class="d1">25</td><td class="d2">152</td><td class="d2">50</td><td><strong>304</strong></td><td><strong>100</strong></td><td class="d5">101</td><td class="d3">33</td><td><strong>202</strong></td></tr>
	</table>		</table>

Line 98:		Line 98:
	: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.		: When this file is displayed in a browser, the zoom factor may be reduced (with Ctrl "-", to 25 % for example) such that the structure of the lengths of streets can be seen.
	====Street lengths====		====Street lengths====
	* The lengths ~~r<sub>1</sub>~~ of the streets seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the street.		* The lengths of the streets seem to be finite. The highlighted pairs of numbers ≡ 4 mod 6 are decreasing to the end of the street.
	* The lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the street construction rule.		* The lengths show a repeating pattern for the start values mod 54. The fixed lengths 3, 4, 5 can probably be explained from the street construction rule.
	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"