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A267062 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones. 8
212, -5760, -165852, -10501476, -449827456, -21948311748, -1016699956620, -48023357086272, -2251419462422716, -105852417560435076, -4971310326775823808, -233572686675369390180, -10972461323000994899692, -515480788238950647507456, -24216468853316695676874396 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

See A265762 for a guide to related sequences.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..595

Index entries for linear recurrences with constant coefficients, signature (34, 714, -4641, -12376, 12376, 4641, -714, -34, 1).

FORMULA

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).

G.f.:  (4 (-53 + 3242 x + 30345 x^2 - 58506 x^3 - 383152 x^4 - 61754 x^5 + 46551 x^6 - 1122 x^7 + 9 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9).

EXAMPLE

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

[u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = 212.

[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -5760;

[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -165852.

MATHEMATICA

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];

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

t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];

Coefficient[t, x, 0];  (* A266803 *)

Coefficient[t, x, 1];  (* A266808 *)

Coefficient[t, x, 2];  (* A267061 *)

Coefficient[t, x, 3];  (* A267062 *)

Coefficient[t, x, 4];  (* A267063 *)

Coefficient[t, x, 5];  (* A267064 *)

Coefficient[t, x, 6];  (* A267065 *)

Coefficient[t, x, 7];  (* A267066 *)

Coefficient[t, x, 8];  (* A266803 *)

CROSSREFS

Cf. A265762, A266803, A266808, A267061, A267063, A267064, A267065, A267066.

Sequence in context: A232671 A082828 A254704 * A250326 A344423 A238023

Adjacent sequences:  A267059 A267060 A267061 * A267063 A267064 A267065

KEYWORD

sign,easy

AUTHOR

Clark Kimberling, Jan 10 2016

STATUS

approved

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Last modified July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)