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A266804 Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones. 5
19, 19, 361, 1795, 14011, 91489, 638899, 4348051, 29883145, 204609571, 1402971259, 9614651329, 65903614291, 451700107795, 3096024736681, 21220400800579, 145446970016059, 996907894114081, 6832909585226995, 46833455808339091, 321001289959109449 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

See A265762 for a guide to related sequences.

LINKS

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

FORMULA

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) .

G.f.:  (-19 + 76 x + 19 x^2 + 10 x^3 - x^4)/(-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(6),1,1,1,...] has p(0,x)=19-14x-13x^2+2x^3+x^4, so a(0) = 19;

[1,sqrt(6),1,1,1,...] has p(1,x)=19-90x+143x^2-90x^3+19x^4, so a(1) = 19;

[1,1,sqrt(6),1,1,1...] has p(2,x)=361-722x+527x^2-166x^3+19x^4, so a(2) = 361.

MATHEMATICA

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}];

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

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

Coefficient[t, x, 0] ; (* A266804 *)

Coefficient[t, x, 1];  (* A266805 *)

Coefficient[t, x, 2];  (* A266806 *)

Coefficient[t, x, 3];  (* A266807 *)

Coefficient[t, x, 4];  (* A266804 *)

CROSSREFS

Cf. A265762, A266805, A266806, A266807.

Sequence in context: A261756 A070853 A165840 * A170914 A103418 A004508

Adjacent sequences:  A266801 A266802 A266803 * A266805 A266806 A266807

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 09 2016

STATUS

approved

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Last modified October 18 17:15 EDT 2021. Contains 348068 sequences. (Running on oeis4.)