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 A329494 #36 Apr 04 2024 04:32:04
%S A329494 1,2,5,14,3,22,13,10,17,38,7,46,25,18,29,62,11,70,37,26,41,86,15,94,
%T A329494 49,34,53,110,19,118,61,42,65,134,23,142,73,50,77,158,27,166,85,58,89,
%U A329494 182,31,190,97,66,101,206,35,214,109,74,113,230,39,238,121
%N A329494 Numerator of 2*(2*n+1)/(n+2).
%C A329494 Also the numerator of Catalan(n+1)/Catalan(n). For denominators - in both cases - see A060789.
%C A329494 The GCD of 2(2n+1) and n+2 is the GCD of 2*3 and n+2, i.e. a(n) = 2n+1 if 2|n+2, a(n) = 2(2n+1)/3 if 3|n+2, and a(n)=(2n+1)/3 if 6|n+2, and a(n)=2(2n+1) otherwise. - _R. J. Mathar_, Jan 09 2020
%H A329494 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2,0,0,0,0,0,-1).
%F A329494 From _Colin Barker_, Nov 17 2019: (Start)
%F A329494 G.f.: (1 + 5*x^2 + x^4)*(1 + 2*x + 4*x^3 + 2*x^4 + x^6 + 2*x^7) / (1 - 2*x^6 + x^12).
%F A329494 a(n) = 2*a(n-6) - a(n-12) for n>11. (End)
%F A329494 a(n) = -(1/18)*(1 + 2*n)*(-21 + 2*cos(n*Pi/3) - 6*cos(2*n*Pi/3) + 7*(-1)^n + 2*sqrt(3)*sin(n*Pi/3) + 6*sqrt(3)*sin(2*n*Pi/3)). - _Stefano Spezia_, Nov 17 2019 after _Colin Barker_
%F A329494 Sum_{k=1..n} a(k) ~ (7/6) * n^2. - _Amiram Eldar_, Apr 04 2024
%e A329494 Fractions begin with 1, 2, 5/2, 14/5, 3, 22/7, 13/4, 10/3, 17/5, 38/11, 7/2, ...
%t A329494 Numerator@Table[(2 (2 n + 1) / (n + 2)), {n, 0, 60}] (* _Vincenzo Librandi_, Jan 10 2020 *)
%o A329494 (PARI) a(n) = numerator(2*(2*n+1)/(n+2)); \\ _Michel Marcus_, Nov 17 2019
%o A329494 (Magma) [Numerator(2*(2*n+1)/(n+2)): n in [0..60]]; // _Vincenzo Librandi_, Jan 10 2020
%Y A329494 Cf. A000108, A060789.
%K A329494 nonn,easy,frac
%O A329494 0,2
%A A329494 _N. J. A. Sloane_, Nov 17 2019 following a suggestion from _Gary W. Adamson_