A290775 Number of 5-cycles in the n-triangular honeycomb bishop graph.

0, 0, 2, 24, 138, 532, 1596, 4032, 8988, 18216, 34254, 60632, 102102, 164892, 256984, 388416, 571608, 821712, 1156986, 1599192, 2174018, 2911524, 3846612, 5019520, 6476340, 8269560, 10458630, 13110552, 16300494, 20112428, 24639792, 29986176, 36266032, 43605408, 52142706

Offset: 1

Author

Eric W. Weisstein, Aug 10 2017

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A034827 (3-cycles in the triangular honeycomb bishop graph), A051843 (4-cycles), A290779 (6-cycles).

Programs

Mathematica

Table[2/5 Binomial[n + 1, 4] (8 - 7 n + 2 n^2), {n, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 2, 24, 138, 532, 1596}, 20]
CoefficientList[Series[-((2 (x^2 + 5 x^3 + 6 x^4))/(-1 + x)^7), {x, 0, 20}], x]

PARI

a(n)=n*(2*n^5 - 11*n^4 + 20*n^3 - 5*n^2 - 22*n + 16)/60 \\ Charles R Greathouse IV, Aug 10 2017

Formula

a(n) = 2/5 * binomial(n + 1, 4)*(8 - 7*n + 2*n^2).

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

G.f.: -((2 x (x^2 + 5 x^3 + 6 x^4))/(-1 + x)^7).