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%I #27 Jan 30 2023 07:39:37
%S 1,2,5,14,42,132,430,1444,4981,17594,63441,232806,866870,3266460,
%T 12426210,47629020,183638729,711285170,2764753405,10775740030,
%U 42086252770,164635420788,644811687734,2527808259668,9916569410301,38923511495402,152841133694345,600349070362454
%N Sequence generated from the E6 Cartan matrix.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/E6_%28mathematics%29">E6 (mathematics)</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-55,120,-125,52,-3).
%F 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.
%F 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
%F a(n) ~ c*(2 + sqrt(2 + sqrt(3)))^n, where c = (3 - sqrt(3))/24. - _Stefano Spezia_, Jan 29 2023
%F 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
%t 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 *)
%o (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
%Y Cf. A126566, A126568, A126569.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Dec 28 2006
%E More terms from _Robert G. Wilson v_, Aug 07 2007
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