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

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

Author

Wolfdieter Lang, Nov 27 2017

Keywords

• nonn
• frac
• cofr
• easy

Comments

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.

Links

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

Crossrefs

Cf. A294969, A295336.

Programs

Mathematica

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

Formula

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).