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A126567
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Sequence generated from the E6 Cartan matrix.
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2
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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