A294969 -id:A294969 - cp's OEIS frontend

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

Wolfdieter Lang, Nov 27 2017

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.

			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, ...

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

Cf. A294969, A295337.

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 *)

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).

A295337 Denominators of the convergents to sqrt(14)/2 = A294969.

Original entry on oeis.org

1, 1, 7, 8, 23, 31, 209, 240, 689, 929, 6263, 7192, 20647, 27839, 187681, 215520, 618721, 834241, 5624167, 6458408, 18540983, 24999391, 168537329, 193536720, 555610769, 749147489, 5050495703, 5799643192, 16649782087

Offset: 0

Wolfdieter Lang, Nov 27 2017

The corresponding numerators are given in A295336.
The recurrence is a(n) = b(n)*a(n-1) + a(n-2), n >= 0, with a(0) = 1, a(-1) = 0, (a(-2) = 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 =  2*x*G_3 + 1 + x*G_2, G_1= G_0 +  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.

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

Cf. A294969, A295336.

Denominator[Convergents[Sqrt[14]/2, 50]] (* Vaclav Kotesovec, Nov 29 2017 *)

G.f.: (1 + x - x^2)*(1 + 8*x^2 + x^4)/(1- 30*x^4 + x^8).

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

cp's OEIS Frontend

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

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