OEIS/coors: Difference between revisions
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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.: | O.g.f.: (-x^13-2*x^12+3*x^10+5*x^9+8*x^8+11*x^7+13*x^6+14*x^5+12*x^4+9*x^3+7*x^2+4*x+1) / (x^10-2*x^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.: | O.g.f.: (x^10+6*x^9+6*x^8+9*x^7+12*x^6+16*x^5+12*x^4+9*x^3+6*x^2+6*x+1) / (x^10-2*x^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 | 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 | 1320 x^4+1 -> y+1 | ||
1061 x^4-x^2+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 | 120 x^6-x^3+1 -> y^2-y+1 | ||
1448 x^6+x^3+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 | 159 x^8+1 -> y+1 | ||
23 x^8-x^4+1 | 23 x^8-x^4+1 -> y^2-y+1 | ||
54 x^8-x^6+x^4-x^2+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 | 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 | 6 x^16+1 -> y+1 | ||
6 x^16-x^12+x^8-x^4+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 | 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 | 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 |
Revision as of 20:34, 13 May 2020
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 example:
2-Uniform Tiling 13 of 20 Standard notation: [3^6; 3^2.4.12] A301692 Gal.2.13.1 O.g.f.: (-x^13-2*x^12+3*x^10+5*x^9+8*x^8+11*x^7+13*x^6+14*x^5+12*x^4+9*x^3+7*x^2+4*x+1) / (x^10-2*x^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 Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2 A301690 Gal.2.13.2 O.g.f.: (x^10+6*x^9+6*x^8+9*x^7+12*x^6+16*x^5+12*x^4+9*x^3+6*x^2+6*x+1) / (x^10-2*x^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 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 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 denominator coefficient lists are always symmetrical.
- 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 42 s5 =s5 147 s7 =s7 241 s9 =s9 539 s11=s11 589 s13=s13 199 s15=s15 114 s17=s17 160 s19=s19 128 s21=s21 110 s23=s23 96 s25=s25 82 s27=s27 85 s29=s29 24 s31=s31 16 s33=s33 33 s35=s35 4 s37=s37
Factors of the denominator polynomials
When the denominator polynomials are factorized, there are only 34 different factors (the leading numbers give the occurrences of the factor):
6536 x-1 2208 x+1 2286 x^2+1 -> y+1 1635 x^2-x+1 3145 x^2+x+1 1320 x^4+1 -> y+1 1061 x^4-x^2+1 -> y^2-y+1 1246 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 1448 x^6+x^3+1 -> y^2+y+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+1 159 x^8+1 -> 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+1 328 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^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 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 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+1 452 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-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 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 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^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
We observe the following properties of for the denominators:
- All denominators have a factor (x-1)^2.
- All coefficents of x in the factors are +1 or -1.
- Except for the factors x-1 and x+1, all factors are of even degree.
- 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.