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A267082 Coefficient of x^4 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones. 7
0, 6, 456, 6240, 131238, 2238780, 41011296, 730283034, 13143304440, 235581102912, 4229156006790, 75876624195564, 1361636473680576, 24432987781993530, 438436202143461288, 7867390833380267040, 141174789462751501926, 2533277512666920359964 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

See A265762 for a guide to related sequences.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (14, 90, -350, 90, 14, -1).

FORMULA

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.

G.f.:  (6 (x + 62 x^2 - 114 x^3 + 823 x^4 - 182 x^5 - 28 x^6 + 2 x^7))/(1 - 14 x - 90 x^2 + 350 x^3 - 90 x^4 - 14 x^5 + x^6).

From Andrew Howroyd, Mar 07 2018: (Start)

a(n) = 14*a(n-1) + 90*a(n-2) - 350*a(n-3) + 90*a(n-4) + 14*a(n-5) - a(n-6) for n > 7.

G.f.: 6*x*(1 + 62*x - 114*x^2 + 823*x^3 - 182*x^4 - 28*x^5 + 2*x^6)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)).

(End)

EXAMPLE

Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:

[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = 0.

[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 6;

[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = 456.

MATHEMATICA

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];

f[n_] := FromContinuedFraction[t[n]];

t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]

Coefficient[t, x, 0]; (* A267078 *)

Coefficient[t, x, 1]; (* A267079 *)

Coefficient[t, x, 2]; (* A267080 *)

Coefficient[t, x, 3]; (* A267081 *)

Coefficient[t, x, 4]; (* A267082 *)

Coefficient[t, x, 5]; (* A267083 *)

Coefficient[t, x, 6]; (* A266527 *)

PROG

(PARI) concat([0], 6*Vec((1 + 62*x - 114*x^2 + 823*x^3 - 182*x^4 - 28*x^5 + 2*x^6)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30))) \\ Andrew Howroyd, Mar 07 2018

CROSSREFS

Cf. A265762, A267078, A267079, A267080, A267081, A267083, A266527.

Sequence in context: A244195 A338943 A232593 * A051735 A024084 A332146

Adjacent sequences:  A267079 A267080 A267081 * A267083 A267084 A267085

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 11 2016

STATUS

approved

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Last modified August 17 02:42 EDT 2022. Contains 356180 sequences. (Running on oeis4.)