A101501 Number of walks between adjacent nodes on C_5 tensor J_2.

0, 1, 0, 12, 8, 160, 224, 2240, 4608, 32512, 84480, 485376, 1464320, 7401472, 24608768, 114606080, 406093824, 1793720320, 6626869248, 28280881152, 107384668160, 448110002176, 1732341923840, 7123849183232, 27866041417728

Offset: 0

Author

Paul Barry, Dec 04 2004

Keywords

• nonn
• easy

Comments

Let (C_5 tensor J_2) be the 10 node graph whose adjacency matrix is the tensor product of that of C_5 and J_2=[1,1;1,1]. Then a(n) counts walks of length n between adjacent vertices of this graph.

References

• E.R. van Dam, Graphs with few eigenvalues, Tilburg, 1968, p53.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,12,-16).

Crossrefs

Cf. A101502.

Programs

Mathematica

LinearRecurrence[{2,12,-16},{0,1,0},30] (* Harvey P. Dale, Apr 09 2022 *)

Formula

G.f.: x(1-2x)/((1-4x)(1+2x-4x^2)); a(n)=2a(n-1)+12a(n-2)-16a(n-3); a(n)=(sqrt(5)-1)^(n+1)/20-(sqrt(5)+1)^(n+1)(-1)^n/20+4^n/10; a(n)=sum{k=0..n, sqrt(5)((sqrt(5)-1)^k/10-(-sqrt(5)-1)^k/10)(4^(n-k)+0^(n-k))/2}.

(1/10) [4^n - (-2)^n*Lucas(n+1) ]. - Ralf Stephan, May 16 2007

a(n) = 2^(n-1)*A052964(n-1). - R. J. Mathar, Mar 08 2021