A126569 Top-left "head" entry of the n-th power of the E8 Cartan matrix.

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63442, 232828, 867146, 3269060, 12446684, 47771496, 184544427, 716658870, 2794956099, 10938266562, 42930256917, 168890693650, 665739119129, 2628578437646, 10393091551794, 41141896235012, 163028816478833

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Wikipedia, E8

Index entries for linear recurrences with constant coefficients, signature (16,-105,364,-714,784,-440,96,-1).

Formula

a(n) = leftmost term in M^n * [1,0,0,0,0,0,0,0], where M = the 8x8 matrix [2,-1,0,0,0,0,0,0; -1,2,-1,0,0,0,0,0; 0,-1,2,-1,0,0,0,-1; 0,0,-1,2,-1,0,0,0; 0,0,0,-1,2,-1,0,0; 0,0,0,0,-1,2,-1,0; 0,0,0,0,0,-1,2,0; 0,0,-1,0,0,0,0,2].

a(n) = 16*a(n-1)-105*a(n-2)+364*a(n-3)-714*a(n-4)+784*a(n-5)-440*a(n-6)+96*a(n-7) -a(n-8). - R. J. Mathar, May 08 2009 [Corrected by Georg Fischer, Mar 12 2020]

G.f.: -(2*x-1)*(2*x^2-4*x+1)*(x^4-16*x^3+20*x^2-8*x+1) / (1-16*x +105*x^2 -364*x^3+714*x^4-784*x^5+440*x^6-96*x^7+x^8). - R. J. Mathar, May 08 2009

Example

a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0,0].

Maple

E8 := matrix(8, 8, [ [2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, -1 ], [ 0, 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, 0, -1, 2, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 2 ] ]) ;
printf("1, ") ; for n from 1 to 20 do T := evalm(E8^n) ; printf("%a, ", T[1, 1]) ; od: # R. J. Mathar, May 08 2009

Prog

(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1; -1, 96, -440, 784, -714, 364, -105, 16]^n*[1; 2; 5; 14; 42; 132; 430; 1444])[1, 1] \\ Charles R Greathouse IV, May 27 2026

Crossrefs

Cf. A126566, A126567, A126568.

Sequence in context: A057413 A126567 A125501 * A162748 A374548 A393659

Adjacent sequences: A126566 A126567 A126568 * A126570 A126571 A126572

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 28 2006

Extensions

Edited by R. J. Mathar, May 08 2009

Status

approved