A027626 Denominator of n*(n+5)/((n+2)*(n+3)).

1, 2, 10, 5, 7, 28, 12, 15, 55, 22, 26, 91, 35, 40, 136, 51, 57, 190, 70, 77, 253, 92, 100, 325, 117, 126, 406, 145, 155, 496, 176, 187, 595, 210, 222, 703, 247, 260, 820, 287, 301, 946, 330, 345, 1081, 376, 392, 1225, 425, 442, 1378, 477, 495, 1540, 532, 551

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Peter Bala, A note on the sequence of numerators of a rational function, 2019.
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A000292, A027625, A061347, A107711, A234041.

Programs

Magma

[Denominator(n*(n+5)/((n+2)*(n+3))): n in [0..60]]; // Vincenzo Librandi, Mar 04 2014

Mathematica

CoefficientList[Series[(1+2*x+10*x^2+2*x^3+x^4-2*x^5+x^8)/(1-x^3)^3, {x,0,50}], x] (* Vincenzo Librandi, Mar 04 2014 *)

PARI

a(n) = numerator((n+2)*(n+3)/6); \\ Altug Alkan, Apr 18 2018

SageMath

[numerator(binomial(n+3,2)/3) for n in (0..60)] # G. C. Greubel, Aug 04 2022

Formula

a(n) = GCD of n-th and (n+1)st tetrahedral numbers (A000292). - Ross La Haye, Sep 13 2003

G.f.: (1 +2*x +10*x^2 +2*x^3 +x^4 -2*x^5 +x^8)/(1-x^3)^3.

a(n) = A234041(n+1) = A107711(n+4,3) = C(n+3,2)*gcd(n+4,3)/3 for n >= 0. See the o.g.f. of A234041. - Wolfdieter Lang, Feb 26 2014

a(n) = numerator of (n+2)*(n+3)/6. - Altug Alkan, Apr 18 2018

Sum_{n>=0} 1/a(n) = 5 - 4*Pi/(3*sqrt(3)). - Amiram Eldar, Aug 11 2022

a(n) = (n + 2)*(n + 3)*(5 - 2*A061347(n+1))/18. - Stefano Spezia, Oct 16 2023

a(n) is quasi-polynomial in n: a(3*n) = (n+1)*(3*n+2)/2 = A000326(n+1); a(3*n+1) = (n+1)*(3*n+4)/2 = A005449(n+1); a(3*n+2) = (3*n+4)*(3*n+5)/2 = A060544(n+2). - Peter Bala, Nov 20 2024

Extensions

More terms from Vincenzo Librandi, Mar 04 2014