A027625 Numerator of n*(n+5)/((n+2)*(n+3)).

0, 1, 7, 4, 6, 25, 11, 14, 52, 21, 25, 88, 34, 39, 133, 50, 56, 187, 69, 76, 250, 91, 99, 322, 116, 125, 403, 144, 154, 493, 175, 186, 592, 209, 221, 700, 246, 259, 817, 286, 300, 943, 329, 344, 1078, 375, 391, 1222

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A027626 (denominator), A095794, A115067, A179436.

Programs

Magma

[Numerator(n*(n+5)/((n+2)*(n+3))): n in [0..50]]; // Vincenzo Librandi, Mar 04 2014

Mathematica

CoefficientList[Series[x*(1+7*x+4*x^2+3*x^3+4*x^4-x^5-x^6-2*x^7)/(1-x^3)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 04 2014 *)
Numerator[25*Binomial[Range[0, 50]/5 +1, 2]/3] (* G. C. Greubel, Aug 05 2022 *)

PARI

a(n) = numerator(n*(n+5)/6); \\ Altug Alkan, Apr 18 2018

SageMath

[numerator(n*(n+5)/6) for n in (0..50)] # G. C. Greubel, Aug 05 2022

Formula

G.f.: x*(1 + 7*x + 4*x^2 + 3*x^3 + 4*x^4 - x^5 - x^6 - 2*x^7)/(1 - x^3)^3.

a(n) = numerator of n*(n+5)/6. - Altug Alkan, Apr 18 2018

From Peter Bala, Aug 06 2022: (Start)

a(n) is quasi-polynomial in n:

a(3*n) = (1/2)*n*(3*n+5) = A115067(n+1).

a(3*n+1) = (1/2)*(n+2)*(3*n+1) = A095794(n+1).

a(3*n+2) = (1/2)*(3*n+2)*(3*n+7) = A179436(n). (End)

Sum_{n>=1} 1/a(n) = 4*Pi/(15*sqrt(3)) + 87/50. - Amiram Eldar, Aug 11 2022