1,2

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.

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

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). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2008]

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].

Sequence in context: A009572 A027202 A026934 this_sequence A079027 A081105 A161727

Adjacent sequences: A131432 A131433 A131434 this_sequence A131436 A131437 A131438

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 11 2007