A364619 Number of 4-cycles in the n-Pell graph.

0, 0, 1, 8, 40, 164, 601, 2048, 6632, 20680, 62633, 185352, 538272, 1538892, 4341905, 12112960, 33464240, 91666192, 249215921, 673049800, 1806888568, 4824913652, 12821690281, 33922774464, 89391291480, 234694621656, 614106591769, 1601882815304, 4166439039664

Offset: 0

Author

Eric W. Weisstein, Jul 30 2023

Keywords

• nonn

References

• E. Munarini, Pell Graphs, Disc. Math., 342 (2019), 2415-2428.

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Pell Graph
• Index entries for linear recurrences with constant coefficients, signature (6,-9,-4,9,6,1).

Programs

Mathematica

Table[(n (2 n - 3) (-I)^n ChebyshevT[n, I] + (2 n^2 - 2 n + 1) Fibonacci[n, 2])/16, {n, 0, 20}]
LinearRecurrence[{6, -9, -4, 9, 6, 1}, {0, 0, 1, 8, 40, 164}, 20]
CoefficientList[Series[-x^2 (1 + x)^2/(-1 + 2 x + x^2)^3, {x, 0, 20}], x]

Formula

a(n) = (n*(2*n - 3)*A001333(n) + (2*n^2 - 2*n + 1)*A000129(n))/16.

G.f.: -x^2*(1+x)^2/(-1+2*x+x^2)^3.

a(n) = 6*a(n-1)-9*a(n-2)-4*a(n-3)+9*a(n-4)+6*a(n-5)+a(n-6).