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A266713
Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.
5
2, 2, -2, -54, -226, -1958, -12382, -87618, -593374, -4085846, -27955618, -191739462, -1313864638, -9006244994, -61727410366, -423092015478, -2899899974242, -19876251587558, -136233746512414, -933760274094786, -6400087386491038, -43866853488227222
OFFSET
0,1
COMMENTS
See A265762 for a guide to related sequences.
FORMULA
a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: -(2*(1 -4*x -21*x^2 -22*x^3 +57*x^4 -20*x^5 -12*x^6 +2*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5).
EXAMPLE
Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = 2;
[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = 2;
[1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = -2.
MATHEMATICA
u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266710 *)
Coefficient[t, x, 1]; (* A266711 *)
Coefficient[t, x, 2]; (* A266712 *)
Coefficient[t, x, 3]; (* A266713 *)
Coefficient[t, x, 4]; (* A266710 *)
LinearRecurrence[{5, 15, -15, -5, 1}, {2, 2, -2, -54, -226, -1958, -12382, -87618}, 30] (* G. C. Greubel, Jan 26 2018 *)
PROG
(PARI) x='x+O('x^30); Vec(-(2*(1 -4*x -21*x^2 -22*x^3 +57*x^4 -20*x^5 -12*x^6 +2*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5))) \\ G. C. Greubel, Jan 26 2018
(Magma) I:=[-54, -226, -1958, -12382, -87618]; [2, 2, -2] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 26 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Clark Kimberling, Jan 09 2016
STATUS
approved