%I #17 Feb 28 2020 04:37:41
%S 1,2,4,7,14,30,83,255,807,2482,7399,21518,61752,176385,504181,1445159,
%T 4153716,11960039,34463630,99316022,286133435,824112803,2373059251,
%U 6832536414,19671776119,56638681010,163078362040,469559902129,1352048562017,3893102975595,11209833959312
%N Sequence relating to the benzene ring.
%C 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.
%D Fan Chung and Shlomo Sternberg, "Mathematics and the Buckyball". Fan Chung Graham homepage.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-5,14,0,-4).
%F 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].
%F 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]
%e a(5) = 30 = leftmost term in M^5 * [1,0,0,0,0,0].
%t LinearRecurrence[{5, -6, -5, 14, 0, -4}, {1, 2, 4, 7, 14, 30}, 40] (* _Amiram Eldar_, Feb 28 2020 *)
%Y Cf. A005578.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Jun 14 2006
%E More terms from _Amiram Eldar_, Feb 28 2020