A126567 Sequence generated from the E6 Cartan matrix.

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63441, 232806, 866870, 3266460, 12426210, 47629020, 183638729, 711285170, 2764753405, 10775740030, 42086252770, 164635420788, 644811687734, 2527808259668, 9916569410301, 38923511495402, 152841133694345, 600349070362454

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Wikipedia, E6 (mathematics)

Index entries for linear recurrences with constant coefficients, signature (12,-55,120,-125,52,-3).

Formula

Let M = [2,-1,0,0,0,0; -1,2,-1,0,0,0; 0,-1,2,-1,0,-1; 0,0,-1,2,-1,0; 0,0,0,-1,2,0; 0,0,-1,0,0,2] then a(n) is the upper left term in M^n.

G.f.: -(2*x-1)*(2*x^4-16*x^3+20*x^2-8*x+1) / ((x-1)*(3*x-1)*(x^4-16*x^3+20*x^2-8*x+1)). - Colin Barker, May 25 2013

a(n) ~ c*(2 + sqrt(2 + sqrt(3)))^n, where c = (3 - sqrt(3))/24. - Stefano Spezia, Jan 29 2023

a(n) = (3^n + 1)/4 + ((3 + sqrt(3))*((2 - sqrt(2 - sqrt(3)))^n + (2 + sqrt(2 - sqrt(3)))^n) + (3 - sqrt(3))*((2 - sqrt(2 + sqrt(3)))^n + (2 + sqrt(2 + sqrt(3)))^n))/24. - Vaclav Kotesovec, Jan 30 2023

Mathematica

f[n_] := (MatrixPower[{{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, -1}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 2}}, n].{1, 0, 0, 0, 0, 0})[[1]]; Table[ f@n, {n, 0, 25}] (* Robert G. Wilson v, Aug 07 2007 *)

Prog

(PARI) a(n) = ([2, -1, 0, 0, 0, 0; -1, 2, -1, 0, 0, 0; 0, -1, 2, -1, 0, -1; 0, 0, -1, 2, -1, 0; 0, 0, 0, -1, 2, 0; 0, 0, -1, 0, 0, 2]^n)[1, 1]; \\ Michel Marcus, Jan 30 2023

Crossrefs

Cf. A126566, A126568, A126569.

Sequence in context: A168491 A000108 A057413 * A125501 A126569 A162748

Adjacent sequences: A126564 A126565 A126566 * A126568 A126569 A126570

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 28 2006

Extensions

More terms from Robert G. Wilson v, Aug 07 2007

Status

approved