OEIS/coors

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 tab-separated file with the result.

As an example we examine the two lines (coseq.s) for 'Gal.2.13 in this file:

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^5-1)^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^5-1)^2

The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:

1. 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.
2. The denominator coefficient lists are always symmetrical.
3. 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:

1. All denominators have a factor (x-1)^2.
2. All coefficents of x in the factors are +1 or -1.
3. Except for the factors x-1 and x+1, all factors are of even degree.
4. The number of different patterns could be further reduced (to 20) with substitutions x^(k*m) -> y^k .
5. 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.