A289705 Number of 4-cycles in the n-triangular honeycomb queen graph.

0, 0, 15, 96, 330, 855, 1866, 3624, 6468, 10818, 17193, 26208, 38598, 55209, 77028, 105168, 140904, 185652, 241011, 308736, 390786, 489291, 606606, 745272, 908076, 1098006, 1318317, 1572480, 1864254, 2197629, 2576904, 3006624, 3491664, 4037160, 4648599, 5331744, 6092730

Offset: 1

Author

Eric W. Weisstein, Jul 14 2017

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Index entries for linear recurrences with constant coefficients, signature (4, -4, -4, 10, -4, -4, 4, -1).

Crossrefs

Cf. A105636 (3-cycles), A289706 (5-cycles), A289707 (6-cycles).

Programs

Mathematica

Table[(24 n^5 + 170 n^4 - 660 n^3 + 160 n^2 + 606 n - 165 + (-1)^n (165 - 30 n))/320, {n, 20}]
LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {0, 0, 15, 96, 330, 855, 1866, 3624}, 20]
CoefficientList[Series[-((3 x^2 (-5 - 12 x - 2 x^2 + 7 x^3))/((-1 + x)^6 (1 + x)^2)), {x, 0, 20}], x]

Formula

a(n) = (24*n^5 + 170*n^4 - 660*n^3 + 160*n^2 + 606*n - 165 + (-1)^n*(165 - 30*n))/320.

a(n) = 4*a(n-1)-4*a(n-2)-4*a(n-3)+10*a(n-4)-4*a(n-5)-4*a(n-6)+4*a(n-7)-a(n-8).

G.f.: (-3*x^3*(-5 - 12*x - 2*x^2 + 7*x^3))/((-1 + x)^6*(1 + x)^2).