A265804 Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,5,1,1,1,...], where 1^n means n ones.

1, 19, 29, 95, 229, 619, 1601, 4211, 11005, 28831, 75461, 197579, 517249, 1354195, 3545309, 9281759, 24299941, 63618091, 166554305, 436044851, 1141580221, 2988695839, 7824507269, 20484825995, 53629970689, 140405086099, 367585287581, 962350776671

Offset

0,2

Comments

See A265762 for a guide to related sequences.

Links

Table of n, a(n) for n=0..27.

Index entries for linear recurrences with constant coefficients, signature (2, 2, -1).

Formula

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

G.f.: (1 + 17 x - 11 x^2)/(1 - 2 x - 2 x^2 + x^3).

Example

Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[5,1,1,1,1,...] = (9+sqrt(5))/2 has p(0,x) = 19 - 9 x + x^2, so a(0) = 1;
[1,5,1,1,1,...] = (47-sqrt(5))/38 has p(1,x) = 29 - 47 x + 19 x^2, so a(1) = 19;
[1,1,5,1,1,...] = (105+sqrt(5))/58 has p(2,x) = 5 - 105 x + 29 x^2, so a(2) = 29.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {5}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A265804 *)
Coefficient[t, x, 1] (* A265805 *)
Coefficient[t, x, 2] (* A236804 *)
LinearRecurrence[{2, 2, -1}, {1, 19, 29}, 30] (* Vincenzo Librandi, Jan 06 2016 *)

Prog

(Magma) I:=[1, 19, 29]; [n le 3 select I[n] else 2*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 06 2016
(PARI) Vec((1+17*x-11*x^2)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ Altug Alkan, Jan 07 2016

Crossrefs

Cf. A265762, A265805.

Sequence in context: A181606 A139886 A089724 * A276732 A038542 A269262

Adjacent sequences: A265801 A265802 A265803 * A265805 A265806 A265807

Keyword

nonn

Author

Clark Kimberling, Jan 05 2016

Status

approved