A339118 Number of cycles in the grid graph P_6 X P_n.

15, 275, 5034, 80626, 1222363, 18438929, 279285399, 4237530095, 64300829449, 975566486675, 14800469958185, 224540402345213, 3406558215857382, 51681816786790684, 784078741397570677, 11895467318139343215, 180469294422664219486, 2737947622842077799930

Offset: 2

Author

Seiichi Manyama, Nov 24 2020

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..500
• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Grid Graph

Crossrefs

Cf. A140517, A145401, A231829.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A(n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    return cycles.len()
def A339118(n):
    return A(6, n)
print([A339118(n) for n in range(2, 13)])

Formula

Empirical g.f.: -x^2 * (15 - 325*x + 3889*x^2 - 32204*x^3 + 166496*x^4 - 439661*x^5 + 117553*x^6 + 2506529*x^7 - 5691052*x^8 - 128310*x^9 + 16209330*x^10 - 15148184*x^11 - 17089827*x^12 + 28709449*x^13 + 11141815*x^14 - 27136640*x^15 - 13792528*x^16 + 20876587*x^17 + 15963209*x^18 - 11646759*x^19 - 10681356*x^20 + 3192142*x^21 + 3419602*x^22 - 252986*x^23 - 401310*x^24 - 43774*x^25 + 13852*x^26 + 2950*x^27 - 278*x^28 - 48*x^29 + 4*x^30) / ((-1 + x)^2 * (-1 + 38*x - 580*x^2 + 4945*x^3 - 26274*x^4 + 84913*x^5 - 122213*x^6 - 183068*x^7 + 1124479*x^8 - 1544617*x^9 - 1129508*x^10 + 5346947*x^11 - 3023145*x^12 - 6147688*x^13 + 6904233*x^14 + 3952819*x^15 - 5690282*x^16 - 4144167*x^17 + 3164355*x^18 + 4915006*x^19 - 1267655*x^20 - 3336331*x^21 + 82962*x^22 + 1051157*x^23 + 93428*x^24 - 119962*x^25 - 23089*x^26 + 2688*x^27 + 1368*x^28 - 34*x^29 - 30*x^30 + 2*x^31)). - Vaclav Kotesovec, Dec 09 2020

Extensions

Terms a(13) and beyond from Andrew Howroyd, Dec 08 2020