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A120262
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Sequence relating to the benzene ring.
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0
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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
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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Fan Chung and Shlomo Sternberg, "Mathematics and the Buckyball". Fan Chung Graham homepage.
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LINKS
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FORMULA
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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]
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EXAMPLE
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a(5) = 30 = leftmost term in M^5 * [1,0,0,0,0,0].
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MATHEMATICA
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LinearRecurrence[{5, -6, -5, 14, 0, -4}, {1, 2, 4, 7, 14, 30}, 40] (* Amiram Eldar, Feb 28 2020 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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