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
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
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jul 11 2007
EXTENSIONS
More terms from Michel Marcus, Jan 21 2019
STATUS
approved