Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Apr 11 2022 18:03:02
%S 1,2,13,15,43,58,391,449,1289,1738,11717,13455,38627,52082,351119,
%T 403201,1157521,1560722,10521853,12082575,34687003,46769578,315304471,
%U 362074049,1039452569,1401526618,9448612277,10850138895
%N Numerators of the convergents to sqrt(14)/2 = A294969.
%C The corresponding denominators are given in A295337.
%C 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)}.
%C 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.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,30,0,0,0,-1).
%F 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).
%F a(n) = 30*a(n-4) - a(n-8), n >= 8, with inputs a(0)..a(7).
%e 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, ...
%t Numerator[Convergents[Sqrt[14]/2, 50]] (* _Vaclav Kotesovec_, Nov 29 2017 *)
%t LinearRecurrence[{0,0,0,30,0,0,0,-1},{1,2,13,15,43,58,391,449},50] (* _Harvey P. Dale_, Apr 11 2022 *)
%Y Cf. A294969, A295337.
%K nonn,frac,cofr,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 27 2017