This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295336 #13 Apr 11 2022 18:03:02
%S A295336 1,2,13,15,43,58,391,449,1289,1738,11717,13455,38627,52082,351119,
%T A295336 403201,1157521,1560722,10521853,12082575,34687003,46769578,315304471,
%U A295336 362074049,1039452569,1401526618,9448612277,10850138895
%N A295336 Numerators of the convergents to sqrt(14)/2 = A294969.
%C A295336 The corresponding denominators are given in A295337.
%C A295336 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 A295336 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 A295336 <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 A295336 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 A295336 a(n) = 30*a(n-4) - a(n-8), n >= 8, with inputs a(0)..a(7).
%e A295336 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 A295336 Numerator[Convergents[Sqrt[14]/2, 50]] (* _Vaclav Kotesovec_, Nov 29 2017 *)
%t A295336 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 A295336 Cf. A294969, A295337.
%K A295336 nonn,frac,cofr,easy
%O A295336 0,2
%A A295336 _Wolfdieter Lang_, Nov 27 2017