A289181 Number of 6-cycles in the n X n knight graph.

0, 0, 0, 20, 164, 616, 1348, 2352, 3628, 5176, 6996, 9088, 11452, 14088, 16996, 20176, 23628, 27352, 31348, 35616, 40156, 44968, 50052, 55408, 61036, 66936, 73108, 79552, 86268, 93256, 100516, 108048, 115852, 123928, 132276, 140896, 149788, 158952, 168388, 178096

Offset: 1

Author

Eric W. Weisstein, Jun 27 2017

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Knight Graph
• Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Crossrefs

Cf. A163433 (4-cycles).

Programs

Mathematica

Table[Length[FindCycle[KnightTourGraph[n, n], {6}, All]], {n, 20}]
Table[Piecewise[{{0, n < 4}, {20, n == 4}, {164, n == 5}}, 4 (34 n^2 - 259 n + 484)], {n, 20}]
Join[{0, 0, 0, 20, 164}, LinearRecurrence[{3, -3, 1}, {1036, 408, 52}, {6, 20}]]
CoefficientList[Series[(4 x^3 (-5 - 26 x - 46 x^2 + 7 x^3 + 2 x^4))/(-1 + x)^3, {x, 0, 20}], x]

PARI

a(n)=4*if(n>5,34*n^2 - 259*n + 484, max(36*n-139, 0)) \\ Charles R Greathouse IV, Oct 21 2022

Formula

For n > 5, a(n) = 4 (34 n^2 - 259 n + 484).

G.f.: (4*x^4*(-5 - 26*x - 46*x^2 + 7*x^3 + 2*x^4))/(-1 + x)^3.