A295336 Numerators of the convergents to sqrt(14)/2 = A294969.

1, 2, 13, 15, 43, 58, 391, 449, 1289, 1738, 11717, 13455, 38627, 52082, 351119, 403201, 1157521, 1560722, 10521853, 12082575, 34687003, 46769578, 315304471, 362074049, 1039452569, 1401526618, 9448612277, 10850138895

Offset

0,2

Comments

The corresponding denominators are given in A295337.

The recurrence is a(n) = b(n)*a(n-1) + a(n-2), n >= 1, with a(0) = 1, a(-1) = 1, with b(n) from the continued fraction b = {1,repeat(1, 6, 1, 2)}.

The g.f.s G_j(x) = Sum_{n>=0} a(4*n+j)*x^k, for j=1..4 satisfy (arguments are omitted): G_0 = 1 + 2*x*G_3 + x*G_2, G_1= G_0 + 1 + x*G_3, G_2 = 6*G_1 + G_0, G_3 = G_2 + G_1. After solving for the G_j(x), one finds for G(x) = Sum_{n>=0} a(n)*x^n = Sum_{j=1..4} x^j*G_j(x^4) the o.g.f. given in the formula section.

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,0,30,0,0,0,-1).

Formula

G.f.: (1 + 2*x + 13*x^2 + 15*x^3 + 13*x^4 - 2*x^5 + x^6 - x^7)/(1 - 30*x^4 + x^8).

a(n) = 30*a(n-4) - a(n-8), n >= 8, with inputs a(0)..a(7).

Example

The convergents a(n)/A295337(n) begin: 1, 2, 13/7, 15/8, 43/23, 58/31, 391/209, 449/240, 1289/689, 1738/929, 11717/6263, 13455/7192, 38627/20647, 52082/27839, 351119/187681, 403201/215520, 1157521/618721, 1560722/834241, ...

Mathematica

Numerator[Convergents[Sqrt[14]/2, 50]] (* Vaclav Kotesovec, Nov 29 2017 *)
LinearRecurrence[{0, 0, 0, 30, 0, 0, 0, -1}, {1, 2, 13, 15, 43, 58, 391, 449}, 50] (* Harvey P. Dale, Apr 11 2022 *)

Crossrefs

Cf. A294969, A295337.

Sequence in context: A022116 A041201 A042155 * A042253 A041645 A318999

Adjacent sequences: A295333 A295334 A295335 * A295337 A295338 A295339

Keyword

nonn,frac,cofr,easy

Author

Wolfdieter Lang, Nov 27 2017

Status

approved