A329494 - OEIS

%I #36 Apr 04 2024 04:32:04

%S 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 49,34,53,110,19,118,61,42,65,134,23,142,73,50,77,158,27,166,85,58,89,

%U 182,31,190,97,66,101,206,35,214,109,74,113,230,39,238,121

%N Numerator of 2*(2*n+1)/(n+2).

%C Also the numerator of Catalan(n+1)/Catalan(n). For denominators - in both cases - see A060789.

%C 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 <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 From _Colin Barker_, Nov 17 2019: (Start)

%F 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 a(n) = 2*a(n-6) - a(n-12) for n>11. (End)

%F 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 Sum_{k=1..n} a(k) ~ (7/6) * n^2. - _Amiram Eldar_, Apr 04 2024

%e Fractions begin with 1, 2, 5/2, 14/5, 3, 22/7, 13/4, 10/3, 17/5, 38/11, 7/2, ...

%t Numerator@Table[(2 (2 n + 1) / (n + 2)), {n, 0, 60}] (* _Vincenzo Librandi_, Jan 10 2020 *)

%o (PARI) a(n) = numerator(2*(2*n+1)/(n+2)); \\ _Michel Marcus_, Nov 17 2019

%o (Magma) [Numerator(2*(2*n+1)/(n+2)): n in [0..60]]; // _Vincenzo Librandi_, Jan 10 2020

%Y Cf. A000108, A060789.

%K nonn,easy,frac

%O 0,2

%A _N. J. A. Sloane_, Nov 17 2019 following a suggestion from _Gary W. Adamson_