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.

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

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 factor have an even degree.
4. With substitutions x^(k*m) -> y^k the number of different factor patterns could be further reduced (to 20).
5. After such substitutions, the factors have the form sum(k=0..n: (+1 or -1)^k*x^k), except for one pattern of degree 8 and two patterns of degree 12.