OEIS/A291939 - Revision history

imported>Gfis: 2 x ")" removed

2018-04-11T09:12:33Z

2 x ")" removed

← Older revision		Revision as of 09:12, 11 April 2018
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	# Position the trailing element e[1] = 1 at coordinates (x,y,z) = (0,0,0).		# Position the trailing element e[1] = 1 at coordinates (x,y,z) = (0,0,0).
	# Investigate all e[i] (i > 1):		# Investigate all e[i] (i > 1):
	# for even e[i] "go right" = store e[i] at (e[i-1)].(x+1), e[i-1].y, e[i-1].z),		# for even e[i] "go right" = store e[i] at (e[i-1].(x+1), e[i-1].y, e[i-1].z),
	# for odd e[i] "go down" = store e[i] at (e[i-1)].x, e[i-1].(y+1), e[i-1].z),		# for odd e[i] "go down" = store e[i] at (e[i-1].x, e[i-1].(y+1), e[i-1].z),
	# whenever that position is not occupied by a different number,		# whenever that position is not occupied by a different number,
	# otherwise "go out", i.e. increase the layer z by one for all new elements		# otherwise "go out", i.e. increase the layer z by one for all new elements

imported>Gfis: /* Generating Perl Program */ Github

2018-03-26T15:08:07Z

Generating Perl Program: Github

← Older revision		Revision as of 15:08, 26 March 2018
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	==Generating Perl Program==		==Generating Perl Program==
	[[File:A291939-3d.png\|600px\|right\|First 5 terms of A291939]]		[[File:A291939-3d.png\|600px\|right\|First 5 terms of A291939]]
	After some experimenting it became clear that the description is incomplete, and that it is difficult to develop the sequence with 2-dimensional paper and pencil. The ~~following~~ Perl program generates the sequence, and it has a more detailled description at the beginning:		After some experimenting it became clear that the description is incomplete, and that it is difficult to develop the sequence with 2-dimensional paper and pencil. The Perl program below generates the sequence, and it has a more detailled description at the beginning. The latest version of the program and its auxilliary files for the 3D visualization can be found on a [https://github.com/gfis/fasces/tree/master/oeis/A291939 Github repository].
	<pre>		<pre>
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	} # allocate		} # allocate
	</nowiki></pre>		</nowiki></pre>

	==Input Collatz sequences==		==Input Collatz sequences==
	The program reads the first 10000 Collatz sequences from [https://oeis.org/A070165/a070165.txt a070165.txt]:		The program reads the first 10000 Collatz sequences from [https://oeis.org/A070165/a070165.txt a070165.txt]:

imported>Gfis: usage again

2018-03-26T11:13:22Z

usage again

← Older revision		Revision as of 11:13, 26 March 2018
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	# and the "layer" z is outside.		# and the "layer" z is outside.
	# Process all CSs with increasing starting number coln = 1, 2, 3 ... 10000.		# Process all CSs with increasing starting number coln = 1, 2, 3 ... 10000.
	# Begin at the end of any CS (4 2 1), and proceeding up to		# Begin at the end of any CS (4, 2, 1), and proceeding up to
	# the starting number. Name the elements e(1) = 1, e(2) = 2, e(3) = 4 etc.		# the starting number. Name the elements e[1] = 1, e[2] = 2, e[3] = 4 etc.
	# Position the trailing element e(1) = 1 at coordinates (x,y,z) = (0,0,0).		# Position the trailing element e[1] = 1 at coordinates (x,y,z) = (0,0,0).
	# Investigate all e[i] (i > 1):		# Investigate all e[i] (i > 1):
	# for even e[i] "go right" = store e[i] at (e[i-1)].(x+1), e[i-1].y, e[i-1].z),		# for even e[i] "go right" = store e[i] at (e[i-1)].(x+1), e[i-1].y, e[i-1].z),
	# for odd e[i] "go down" = store e[i] at (e[i-1)].x, e[i-1].(y+1), e[i-1].z),		# for odd e[i] "go down" = store e[i] at (e[i-1)].x, e[i-1].(y+1), e[i-1].z),
	# whenever that position is not occupied by a different number,		# whenever that position is not occupied by a different number,
	# otherwise "go up", i.e. increase the layer z by one for all new elements		# otherwise "go out", i.e. increase the layer z by one for all new elements
	# to be stored from now on.		# to be stored from now on.
	# The target sequence A291939 = a(n) consists of the starting		# The target sequence A291939 = a(n) consists of the starting
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	* a fixed frame '''<code>[http://www.teherba.org/threejs/a291939-3d.html a291939-3d.html]</code>''' for the display of the sequence which is shown below.		* a fixed frame '''<code>[http://www.teherba.org/threejs/a291939-3d.html a291939-3d.html]</code>''' for the display of the sequence which is shown below.

	Once that HTML file with the Javascript functions has been loaded (that may take about 15 s), the generated 3D structure is shown as a sequence of colored cubes, with the Collatz sequence elements written inside. Each layer has a different color, and the beginnings of the layers (which form sequence A291939) are ~~labelled~~ with the corresponding ~~element~~. The structure can be rotated by clicking and moving the mouse, ~~and zooming is done~~ by turning the mouse wheel~~. Shift~~ up/down/left/right ~~is done~~ by the cursor arrow keys.		Once that HTML file with the Javascript functions has been loaded (that may take about 15 s), the generated 3D structure is shown as a sequence of colored cubes, with the Collatz sequence elements written inside. The interesting part is the region with red elements in the upper left. Each layer has a different color, and the beginnings of the layers (which form sequence A291939) are highlighted with white edges and a pointer to the corresponding CS sequence number (1, 12, 19, 27, 37 ...).
			The structure can be
			* rotated by clicking and moving the mouse,
			* zoomed by turning the mouse wheel,
			* shifted up, down, left and right by the cursor arrow keys.
			Only the first 100 Collatz sequences were processed for this example (parameter <code>SIZE</code> in the <code>makefile</code>). The resulting structure is already rather extended because of the CS sequences with more than 100 elements (for example 27).

	~~Only~~ the ~~first 100 Collatz sequences were processed for this example (parameter SIZE in~~ the <code>~~makefile~~</code>~~), since more data would result in longer loading times and more tedious scrolling on the display~~.		The HTML file must be stored on a webserver together with the Javascripts from the [http://threejs.org threejs.org] distribution which are referenced at the beginning of the body of <code>[http://www.teherba.org/threejs/a291939-3d.html a291939-3d.html]</code>.
	<pre>		<pre>
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imported>Gfis: SIZE=100

2018-03-25T19:12:43Z

SIZE=100

← Older revision		Revision as of 19:12, 25 March 2018
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	#----------		#----------
	MODE=0		MODE=0
	SIZE=25		SIZE=100
	all: bfile		all: bfile
	#----------		#----------
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	Once that HTML file with the Javascript functions has been loaded (that may take about 15 s), the generated 3D structure is shown as a sequence of colored cubes, with the Collatz sequence elements written inside. Each layer has a different color, and the beginnings of the layers (which form sequence A291939) are labelled with the corresponding element. The structure can be rotated by clicking and moving the mouse, and zooming is done by turning the mouse wheel. Shift up/down/left/right is done by the cursor arrow keys.		Once that HTML file with the Javascript functions has been loaded (that may take about 15 s), the generated 3D structure is shown as a sequence of colored cubes, with the Collatz sequence elements written inside. Each layer has a different color, and the beginnings of the layers (which form sequence A291939) are labelled with the corresponding element. The structure can be rotated by clicking and moving the mouse, and zooming is done by turning the mouse wheel. Shift up/down/left/right is done by the cursor arrow keys.

			Only the first 100 Collatz sequences were processed for this example (parameter SIZE in the <code>makefile</code>), since more data would result in longer loading times and more tedious scrolling on the display.
	<pre>		<pre>
	<nowiki>		<nowiki>

imported>Gfis at 19:02, 25 March 2018

2018-03-25T19:02:59Z

← Older revision		Revision as of 19:02, 25 March 2018
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	</pre>		</pre>
	== 3D Visualization ==		== 3D Visualization ==
	For the visualization, the Javascript library '''[http://threejs.org Three.js]''' is used. There is		For the visualization, the Javascript library '''[http://threejs.org Three.js]''' is used. There is
	* a variable data file <code>a291939-3d.js</code> which is generated by the makefile, and		* a variable data file <code>a291939-3d.js</code> which is generated by the <code>makefile</code> above, and
	* a fixed frame '''<code>[http://www.teherba.org/threejs/a291939-3d.html a291939-3d.html]</code>''' for the display of the sequence which is shown below.		* a fixed frame '''<code>[http://www.teherba.org/threejs/a291939-3d.html a291939-3d.html]</code>''' for the display of the sequence which is shown below.

	Once ~~the~~ HTML file with the Javascript functions has been loaded (that may take about 15 s), the generated 3D structure is shown as a sequence of colored cubes, with the Collatz sequence elements written inside. Each layer has a different color, and the beginnings of the layers (which form sequence A291939) are labelled with the corresponding element. The structure can be rotated by clicking and moving the mouse, and zooming is done by turning the mouse wheel. Shift up/down/left/right is done by the cursor arrow keys.		Once that HTML file with the Javascript functions has been loaded (that may take about 15 s), the generated 3D structure is shown as a sequence of colored cubes, with the Collatz sequence elements written inside. Each layer has a different color, and the beginnings of the layers (which form sequence A291939) are labelled with the corresponding element. The structure can be rotated by clicking and moving the mouse, and zooming is done by turning the mouse wheel. Shift up/down/left/right is done by the cursor arrow keys.
	<pre>		<pre>
	<nowiki>		<nowiki>

imported>Gfis: altogether

2018-03-25T19:01:29Z

altogether

Show changes

imported>Gfis: with .pl

2018-03-25T18:38:07Z

with .pl

Show changes

imported>Gfis: Created page with "==First number on layer n of hailstone chain== The sequence had keyword `unkn` until March 2018, and there were only 5 terms: * 1, 12, 19, 27, 37 The comment read:..."

2018-03-25T16:28:36Z

Created page with "==First number on layer n of hailstone chain== The sequence had keyword <code>unkn</code> until March 2018, and there were only 5 terms: * 1, 12, 19, 27, 37 The comment read:..."

New page

==First number on layer n of hailstone chain==
The sequence had keyword <code>unkn</code> until March 2018, and there were only 5 terms:
* 1, 12, 19, 27, 37
The comment read:
If hailstone chains are strictly drawn in numerical order at right angles with consistent
direction, overlaps occur. The first set of numbers that do not overlap could be considered the
'first layer'. Once an overlap is needed, all numbers farther up the chain (inclusive) are on a
higher layer. This is the sequence of the first numbers to appear on layer n.
''Hailstone chain'' is a synonym for Collatz sequences or the ''3n+1 problem'' - see [https://oeis.org/index/3#3x1 Index entries for sequences related to 3x+1 (or Collatz) problem].

After some experimenting it became clear that the description is incomplete, and that it is difficult to develop the sequence with 2-dimensional paper and pencil.
==Generating Perl Program==
The following Perl program has a more detailled description at the beginning. The program generates the 5 terms, and 1769
Take the Collatz sequences (CSs) from A070165, and build a 3-dimensional
structure representing all CSs up to some starting number (10000).
In that 3D structure, the y direction is downwards, x is to the right,
and the "layer" z is outside.
Process all CSs with increasing starting number coln = 1, 2, 3 ... 10000.
Begin at the end of any CS (4 2 1), and proceeding up to
the starting number. Name the elements e(1) = 1, e(2) = 2, e(3) = 4 etc.
Position the trailing element e(1) = 1 at coordinates (x,y,z) = (0,0,0).
Investigate all e[i] (i > 1):
for even e[i] "go right" = store e[i] at (e[i-1)].(x+1), e[i-1].y, e[i-1].z),
for odd e[i] "go down" = store e[i] at (e[i-1)].x, e[i-1].(y+1), e[i-1].z),
whenever that position is not occupied by a different number,
otherwise "go up", i.e. increase the layer z by one for all new elements
to be stored from now on.
The target sequence A291939 = a(n) consists of the starting
values of the CSs which reach a z coordinate of n for the first
time.

The is a table of the Collatz sequences up to starting value 10000 in [https://oeis.org/A070165 A070165/a070165.txt].