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A131435
Recursive sequence generated from a Petersen graph.
0
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
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
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jul 11 2007
EXTENSIONS
More terms from Michel Marcus, Jan 21 2019
STATUS
approved