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A120262 Sequence relating to the benzene ring. 0
1, 2, 4, 7, 14, 30, 83, 255, 807, 2482, 7399, 21518, 61752, 176385, 504181, 1445159, 4153716, 11960039, 34463630, 99316022, 286133435, 824112803, 2373059251, 6832536414, 19671776119, 56638681010, 163078362040, 469559902129, 1352048562017, 3893102975595, 11209833959312 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..30.

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

Cf. A005578.

Sequence in context: A157133 A202850 A247295 * A202849 A202842 A013326

Adjacent sequences:  A120259 A120260 A120261 * A120263 A120264 A120265

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, Jun 14 2006

EXTENSIONS

More terms from Amiram Eldar, Feb 28 2020

STATUS

approved

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Last modified September 19 17:35 EDT 2021. Contains 347564 sequences. (Running on oeis4.)