OFFSET
0,2
COMMENTS
The number of cubes reachable by at most n steps across faces in the {4,3,5} tessellation of hyperbolic space, for n >= 0.
LINKS
Eryk Kopczynski, Table of n, a(n) for n = 0..999
Tim Hutton, Generating code in C++, using VTK (gives incorrect terms from some point on!)
Eryk Kopczynski, HyperRogue. Run with parameters -geo 435h -csolve; compile with -DCAP_GMP=0 to get the conjectured formula.
FORMULA
a(d+17) = 3*a(d+16) + 2*a(d+15) + 7*a(d+14) + a(d+13) - 5*a(d+12) + 3*a(d+11) - 2*a(d+10) - 18*a(d+9) + 18*a(d+8) + 2*a(d+7) - 3*a(d+6) + 5*a(d+5) - a(d+4) - 7*a(d+3) - 2*a(d+2) - 3*a(d+1) + a(d) (conjectured, found experimentally and tested from 19 to 135). - Eryk Kopczynski, Jul 04 2020
Conjectured G.f.: (1+x) * (1+2*x+8*x^2+9*x^3+8*x^4+17*x^5+10*x^6+10*x^8+10*x^10+17*x^11+8*x^12+9*x^13+8*x^14+2*x^15+x^16) / ((1-x)^2 * (1-2*x-4*x^2-11*x^3-12*x^4-7*x^5-10*x^6-8*x^7+10*x^8-8*x^9-10*x^10-7*x^11-12*x^12-11*x^13-4*x^14-2*x^15+x^16)). - Natalia L. Skirrow, Apr 29 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Tim Hutton, Sep 11 2014
EXTENSIONS
Offset and terms corrected and more terms added by Eryk Kopczynski, Jul 04 2020
STATUS
approved
