A329494 Numerator of 2*(2*n+1)/(n+2).

1, 2, 5, 14, 3, 22, 13, 10, 17, 38, 7, 46, 25, 18, 29, 62, 11, 70, 37, 26, 41, 86, 15, 94, 49, 34, 53, 110, 19, 118, 61, 42, 65, 134, 23, 142, 73, 50, 77, 158, 27, 166, 85, 58, 89, 182, 31, 190, 97, 66, 101, 206, 35, 214, 109, 74, 113, 230, 39, 238, 121

Offset: 0

Author

N. J. A. Sloane, Nov 17 2019 following a suggestion from Gary W. Adamson

Keywords

• nonn
• easy
• frac

Comments

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

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

Examples

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

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Crossrefs

Cf. A000108, A060789.

Programs

Magma

[Numerator(2*(2*n+1)/(n+2)): n in [0..60]]; // Vincenzo Librandi, Jan 10 2020

Mathematica

Numerator@Table[(2 (2 n + 1) / (n + 2)), {n, 0, 60}] (* Vincenzo Librandi, Jan 10 2020 *)

PARI

a(n) = numerator(2*(2*n+1)/(n+2)); \\ Michel Marcus, Nov 17 2019

Formula

From Colin Barker, Nov 17 2019: (Start)

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).

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

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

Sum_{k=1..n} a(k) ~ (7/6) * n^2. - Amiram Eldar, Apr 04 2024