OEIS/coors - Revision history

Gfis: /* Factors of the denominator polynomials */ writing

2023-09-08T21:36:43Z

Factors of the denominator polynomials: writing

← Older revision		Revision as of 21:36, 8 September 2023
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	# If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences).		# If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences).
	===Factors of the denominator polynomials ===		===Factors of the denominator polynomials ===
	For the denominators of the g.f.s several factorization ~~mathods~~ (Mathematica ''FactorList'' and ''Simplify'', Maple ''factor'') were tried with different results. In the end, a Maple procedure of Robert Israel was used:		For the denominators of the g.f.s several factorization methods (Mathematica's ''FactorList'' and ''Simplify'', Maple's ''factor'') were tried with different results. In the end, a Maple procedure of Robert Israel was used:
	myfactor:= proc(p) local X, k, P, T, q;		myfactor:= proc(p) local X, k, P, T, q;
	P:= p; T:= 1;		P:= p; T:= 1;

imported>Gfis: Reordered

2020-05-19T05:29:14Z

Reordered

← Older revision		Revision as of 05:29, 19 May 2020
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	T * factor(P)		T * factor(P)
	end proc;		end proc;
	With this factorization, the ~~denominators show~~ the following 55 different factors (with number of occurrences prefixed):		With this factorization:
			* Most of the factors have the form ''x^k-1''.
			* There are no coefficients > 1.
			* Some of the polynomials could be further reduced by a substitutions ''x^m -> y'' .
			* After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one polynomial of degree 8 and two polynomials of degree 12.
			There are the following 55 different factors in the denominators (with their number of occurrences prefixed):
	892 x+1		892 x+1
	2091 x-1		2091 x-1
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	6 x^36-1		6 x^36-1
	6 x^40-1		6 x^40-1
	* Most of the factors have the form ''x^k-1''.
	* There are no coefficients > 1.
	* Some of the polynomials could be further reduced by a substitutions ''x^m -> y'' .
	* After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one polynomials of degree 8 and two polynomials of degree 12.
	===See also===		===See also===
	* [[OEIS/Tilings\|Literature and new format]]		* [[OEIS/Tilings\|Literature and new format]]

imported>Gfis: 55 factors

2020-05-19T05:17:52Z

55 factors

Show changes

imported>Gfis: start and end with 1

2020-05-14T17:22:29Z

start and end with 1

← Older revision		Revision as of 17:22, 14 May 2020
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	Denom. factorized: (x^5-1)^2		Denom. factorized: (x^5-1)^2
	The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:		The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:
	# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.		# The coefficient lists for the denominators are always, those for the numerators are often '''symmetrical''' (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.
	# ~~The denominator~~ coefficient lists ~~are always symmetrical~~.		# All coefficient lists start and end with coefficent 1.
	# If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences):		# If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences):
	16 s3 =s3		16 s3 =s3
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	# The number of different patterns could be further reduced (to 20) with substitutions ''x^(k*m) -> y^k'' .		# The number of different patterns could be further reduced (to 20) with substitutions ''x^(k*m) -> y^k'' .
	# After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one pattern of degree 8 and two patterns of degree 12.		# After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one pattern of degree 8 and two patterns of degree 12.

	===See also===		===See also===
	* [[OEIS/Tilings\|Literature and new format]]		* [[OEIS/Tilings\|Literature and new format]]

imported>Gfis: Link to Tilings

2020-05-14T17:18:00Z

Link to Tilings

← Older revision		Revision as of 17:18, 14 May 2020
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	For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.		For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.

	As an example we examine the two lines (coseq.s) for '''Gal.2.13'' in this file:		As an example we examine the two lines (coseq.s) for '''Gal.2.13''' in this file:
	[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]		[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]
	[https://oeis.org/A301692 A301692] Gal.2.13.1		[https://oeis.org/A301692 A301692] Gal.2.13.1
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	# The number of different patterns could be further reduced (to 20) with substitutions ''x^(k*m) -> y^k'' .		# The number of different patterns could be further reduced (to 20) with substitutions ''x^(k*m) -> y^k'' .
	# After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one pattern of degree 8 and two patterns of degree 12.		# After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one pattern of degree 8 and two patterns of degree 12.

			===See also===
			* [[OEIS/Tilings\|Literature and new format]]

imported>Gfis at 13:34, 14 May 2020

2020-05-14T13:34:13Z

imported>Gfis at 13:30, 14 May 2020

2020-05-14T13:30:33Z

imported>Gfis at 13:14, 14 May 2020

2020-05-14T13:14:56Z

← Older revision		Revision as of 13:14, 14 May 2020
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			[[File:Gal.2.13.1.png\|100px\|right\|link=http://www.teherba.org/OEIS-mat/coors/Test.svg]]
	==Generating functions for coordination sequences of uniform tilings==		==Generating functions for coordination sequences of uniform tilings==
	For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.		For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.

imported>Gfis: evalgf.txt

2020-05-13T20:53:19Z

evalgf.txt

← Older revision		Revision as of 20:53, 13 May 2020
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	==Generating functions for coordination sequences of uniform tilings==		==Generating functions for coordination sequences of uniform tilings==
	For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms.		For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.

	~~For~~ example:		As an example we examine two lines (coseq.s) in this file:
	[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]		[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]
	[https://oeis.org/A301692 A301692] Gal.2.13.1		[https://oeis.org/A301692 A301692] Gal.2.13.1

imported>Gfis: factorized

2020-05-13T20:34:15Z

factorized

← Older revision		Revision as of 20:34, 13 May 2020
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	For example:		For example:
			[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]
	[https://oeis.org/A301692 A301692] Gal.2.13.1		[https://oeis.org/A301692 A301692] Gal.2.13.1
	O.g.f.: -(-x^13-2x^12+3x^10+5x^9+8x^8+11x^7+13x^6+14x^5+12x^4+9x^3+7x^2+4x+1)/(-x^10+2x^5-1)		O.g.f.: (-x^13-2x^12+3x^10+5x^9+8x^8+11x^7+13x^6+14x^5+12x^4+9x^3+7x^2+4x+1) / (x^10-2x^5+1)
	Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11		Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11
			Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2
	[https://oeis.org/A301690 A301690] Gal.2.13.2		[https://oeis.org/A301690 A301690] Gal.2.13.2
	O.g.f.: -(x^10+6x^9+6x^8+9x^7+12x^6+16x^5+12x^4+9x^3+6x^2+6x+1)/(-x^10+2x^5-1)		O.g.f.: (x^10+6x^9+6x^8+9x^7+12x^6+16x^5+12x^4+9x^3+6x^2+6x+1) / (x^10-2x^5+1)
	Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11		Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11
			Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2
	The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:		The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:
	# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.		# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.
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	6536 x-1		6536 x-1
	2208 x+1		2208 x+1
	2286 x^2+1 -> y+1		2286 x^2+1 -> y+1
	1635 x^2-x+1		1635 x^2-x+1
	3145 x^2+x+1		3145 x^2+x+1
	1320 x^4+1 -> y+1		1320 x^4+1 -> y+1
	1061 x^4-x^2+1 -> y^2-y+1		1061 x^4-x^2+1 -> y^2-y+1
	1246 x^4-x^3+x^2-x+1		1246 x^4-x^3+x^2-x+1
	2285 x^4+x^3+x^2+x+1		2285 x^4+x^3+x^2+x+1
	120 x^6-x^3+1 -> y^2-y+1		120 x^6-x^3+1 -> y^2-y+1
	1448 x^6+x^3+1 -> y^2+y+1		1448 x^6+x^3+1 -> y^2+y+1
	271 x^6-x^5+x^4-x^3+x^2-x+1		271 x^6-x^5+x^4-x^3+x^2-x+1
	1474 x^6+x^5+x^4+x^3+x^2+x+1		1474 x^6+x^5+x^4+x^3+x^2+x+1
	159 x^8+1 -> y+1		159 x^8+1 -> y+1
	23 x^8-x^4+1 -> y^2-y+1		23 x^8-x^4+1 -> y^2-y+1
	54 x^8-x^6+x^4-x^2+1 -> y^4-y^3+y^2-y+1		54 x^8-x^6+x^4-x^2+1 -> y^4-y^3+y^2-y+1
	328 x^8-x^7 +x^5-x^4+x^3 -x+1		328 x^8-x^7 +x^5-x^4+x^3 -x+1
	6 x^8+x^7 -x^5-x^4-x^3 +x+1		6 x^8+x^7 -x^5-x^4-x^3 +x+1
	6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1		6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
	1492 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1		1492 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
	11 x^12-x^6+1 -> y^2-y+1		11 x^12-x^6+1 -> y^2-y+1
	34 x^12-x^10+x^8-x^6+x^4-x^2+1 -> y^6-y^5+y^4-y^3+y^2-y+1		34 x^12-x^10+x^8-x^6+x^4-x^2+1 -> y^6-y^5+y^4-y^3+y^2-y+1
	6 x^12+x^11 -x^9-x^8 +x^6 -x^4-x^3 +x+1		6 x^12+x^11 -x^9-x^8 +x^6 -x^4-x^3 +x+1
	5 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1		5 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
	32 x^12-x^11 +x^9-x^8 +x^6 -x^4+x^3 -x+1		32 x^12-x^11 +x^9-x^8 +x^6 -x^4+x^3 -x+1
	452 x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1		452 x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
	6 x^16+1 -> y+1		6 x^16+1 -> y+1
	6 x^16-x^12+x^8-x^4+1 -> y^4-y^3+y^2-y+1		6 x^16-x^12+x^8-x^4+1 -> y^4-y^3+y^2-y+1
	89 x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1		89 x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
	12 x^18+x^9+1 -> y^2+y+1		12 x^18+x^9+1 -> y^2+y+1
	59 x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1		59 x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
	18 x^20+x^15+x^10+x^5+1 -> y^4-y^3+y^2-y+1		18 x^20+x^15+x^10+x^5+1 -> y^4-y^3+y^2-y+1
	6 x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1		6 x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
	6 x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1		6 x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1

← Older revision		Revision as of 13:34, 14 May 2020
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	[[File:Gal.2.13.1.png\|~~250px~~\|right\|link=http://www.teherba.org/OEIS-mat/coors/Test.svg]]		[[File:Gal.2.13.1.png\|180px\|right\|link=http://www.teherba.org/OEIS-mat/coors/Test.svg]]
	==Generating functions for coordination sequences of uniform tilings==		==Generating functions for coordination sequences of uniform tilings==
	For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.		For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.
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	O.g.f.: (-x^13-2x^12+3x^10+5x^9+8x^8+11x^7+13x^6+14x^5+12x^4+9x^3+7x^2+4x+1) / (x^10-2x^5+1)		O.g.f.: (-x^13-2x^12+3x^10+5x^9+8x^8+11x^7+13x^6+14x^5+12x^4+9x^3+7x^2+4x+1) / (x^10-2x^5+1)
	Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11		Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11
	Denom. factorized: (x-~~1)^2*(x^4+x^3+x^2+x+~~1)^2		Denom. factorized: (x^5-1)^2
	[https://oeis.org/A301690 A301690] Gal.2.13.2		[https://oeis.org/A301690 A301690] Gal.2.13.2
	O.g.f.: (x^10+6x^9+6x^8+9x^7+12x^6+16x^5+12x^4+9x^3+6x^2+6x+1) / (x^10-2x^5+1)		O.g.f.: (x^10+6x^9+6x^8+9x^7+12x^6+16x^5+12x^4+9x^3+6x^2+6x+1) / (x^10-2x^5+1)
	Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11		Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11
	Denom. factorized: (x-~~1)^2*(x^4+x^3+x^2+x+~~1)^2		Denom. factorized: (x^5-1)^2
	The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:		The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:
	# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.		# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.

← Older revision		Revision as of 13:30, 14 May 2020
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	[[File:Gal.2.13.1.png\|~~100px~~\|right\|link=http://www.teherba.org/OEIS-mat/coors/Test.svg]]		[[File:Gal.2.13.1.png\|250px\|right\|link=http://www.teherba.org/OEIS-mat/coors/Test.svg]]
	==Generating functions for coordination sequences of uniform tilings==		==Generating functions for coordination sequences of uniform tilings==
	For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.		For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.

	As an example we examine two lines (coseq.s) in this file:		As an example we examine the two lines (coseq.s) for '''Gal.2.13'' in this file:
	[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]		[https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]
	[https://oeis.org/A301692 A301692] Gal.2.13.1		[https://oeis.org/A301692 A301692] Gal.2.13.1
	O.g.f.: (-x^13-2x^12+3x^10+5x^9+8x^8+11x^7+13x^6+14x^5+12x^4+9x^3+7x^2+4x+1) / (x^10-2x^5+1)		O.g.f.: (-x^13-2x^12+3x^10+5x^9+8x^8+11x^7+13x^6+14x^5+12x^4+9x^3+7x^2+4x+1) / (x^10-2x^5+1)
	Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11		Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1 a14>s11
	Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2		Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2
	[https://oeis.org/A301690 A301690] Gal.2.13.2		[https://oeis.org/A301690 A301690] Gal.2.13.2
	O.g.f.: (x^10+6x^9+6x^8+9x^7+12x^6+16x^5+12x^4+9x^3+6x^2+6x+1) / (x^10-2x^5+1)		O.g.f.: (x^10+6x^9+6x^8+9x^7+12x^6+16x^5+12x^4+9x^3+6x^2+6x+1) / (x^10-2x^5+1)
	Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11		Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1 s11=s11
	Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2		Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2
	The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:		The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:
	# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.		# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.