A131435 Recursive sequence generated from a Petersen graph.

1, 6, 35, 198, 1124, 6373, 36142, 204959, 1162306, 6591376, 37379241, 211975382, 1202098747, 6817026030, 38658920812, 219232286125, 1243252366462, 7050405210295, 39982400119754, 226737651576696, 1285814820537777, 7291774177355046, 41351188214146259, 234499961894359766

Offset

1,2

Comments

Characteristic polynomial of M = x^5 - 3x^4 - 15x^3 - 3x^2 + 13x - 4. a(n)/a(n-1) tends to 5.6709364838...the largest root of the polynomial and an eigenvalue of the matrix.

Links

Table of n, a(n) for n=1..24.

Stephan G. Wagner, The Fibonacci Number of Generalized Petersen Graphs, Fibonacci Quarterly, Vol. 44, Number 4, November 2006, p. 366.

Formula

Let M = the 5x5 adjacency matrix of a Petersen graph, [Wagner]: [2,1,1,1,0; 1,1,0,1,0; 8,5,0,3,0; 3,2,0,0,1; 5,3,0,3,0]. Then a(n) = M^n (2,1); = second term from the left of M^n * [1,0,0,0,0]. For n>5, a(n) = 3*a(n-1) + 15*a(n-2) + 3*a(n-3) - 13*a(n-4) + 4*a(n-5).

G.f.: x(1+x)(1+2x)/(1-3x-15x^2-3x^3+13x^4-4x^5). [R. J. Mathar, Oct 30 2008]

Example

a(8) = 204959 = 3*36142 + 15*6373 + 3*1124 - 13*198 + 4*35, = 3*a(7) + 15*a(6) + 3*a(5) - 13*a(4) + 4*a(3).
a(5) = 1124 = second term from the left of M^5 * [1,0,0,0,0] = [2669, 1124, 6148, 2580, 4324].

Prog

(PARI) Vec(x*(1+x)*(1+2*x)/(1-3*x-15*x^2-3*x^3+13*x^4-4*x^5) + O(x^30)) \\ Michel Marcus, Jan 21 2019

Crossrefs

Sequence in context: A026934 A171311 A230713 * A209179 A370036 A081105

Adjacent sequences: A131432 A131433 A131434 * A131436 A131437 A131438

Keyword

nonn

Author

Gary W. Adamson, Jul 11 2007

Extensions

More terms from Michel Marcus, Jan 21 2019

Status

approved