A247308 Layer counting sequence in the order-5 cubic honeycomb.

1, 7, 37, 163, 661, 2643, 10497, 41511, 164073, 648495, 2562749, 10127291, 40020845, 158152811, 624980489, 2469769903, 9759926065, 38568829879, 152414547541, 602304889075, 2380161078405, 9405812345187, 37169461719153, 146884589311479, 580451843386809, 2293803210617951, 9064547264192237, 35820865853787467

Offset: 0

Author

Tim Hutton, Sep 11 2014

Keywords

• nonn

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.

Crossrefs

For the {5,3,4} tessellation: A076765.

For the {5,4} tessellation: A054888.

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

Extensions

Offset and terms corrected and more terms added by Eryk Kopczynski, Jul 04 2020