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
Links
- Wikipedia, E6 (mathematics)
- Index entries for linear recurrences with constant coefficients, signature (12,-55,120,-125,52,-3).
Programs
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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 *) -
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
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
Extensions
More terms from Robert G. Wilson v, Aug 07 2007