OFFSET
0,2
COMMENTS
a(n)/a(n-1) tends to the largest eigenvalue of the matrix: (1 + Cos Pi/9) = 2.87938524157... A005578 can be generated by A^n * [1,0,0,0,0,0], leftmost nonzero term.
REFERENCES
Fan Chung and Shlomo Sternberg, "Mathematics and the Buckyball". Fan Chung Graham homepage.
LINKS
Index entries for linear recurrences with constant coefficients, signature (5,-6,-5,14,0,-4).
FORMULA
Let A = the 6x2 adjacency matrix of a benzene ring (reference): [0,1,0,0,0,1; 1,0,1,0,0,0; 0,1,0,1,0,0; 0,0,1,0,1,0; 0,0,0,1,0,1; 1,0,0,0,1,0]. Then perform M = A^2 - A = [2,-1,1,0,1,-1; -1,2,-1,1,0,1; 1,-1,2,-1,1,0; 0,1,-1,2,-1,1; 1,0,1,-1,2,-1; -1,1,0,1,-1,2]. a(n) = leftmost term in M^n * [1,0,0,0,0,0].
G.f.: -(6*x^5+x^4-4*x^3+3*x-1) / ((x^3-3*x+1)*(4*x^3-2*x+1)). [Colin Barker, Nov 29 2012]
EXAMPLE
a(5) = 30 = leftmost term in M^5 * [1,0,0,0,0,0].
MATHEMATICA
LinearRecurrence[{5, -6, -5, 14, 0, -4}, {1, 2, 4, 7, 14, 30}, 40] (* Amiram Eldar, Feb 28 2020 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jun 14 2006
EXTENSIONS
More terms from Amiram Eldar, Feb 28 2020
STATUS
approved