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
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
KEYWORD
nonn,frac,cofr,easy
AUTHOR
Wolfdieter Lang, Nov 27 2017
STATUS
approved