A234041 a(n) = binomial(n+2,2)*gcd(n,3)/3, n >= 0.

1, 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

Offset: 0

Author

Wolfdieter Lang, Feb 24 2014

Keywords

• nonn
• easy

Comments

Apart from the first term, this is the same as A027626. - Bruno Berselli, Feb 24 2014

This is the sequence of the fourth column of the triangle A107711.

Examples

			a(6) = binomial(8,2) = 28 (example for n == 0 (mod 3)),
a(7) = binomial(9,2)/3 = 3*4 = 12 (example for n == 1 (mod 3)),
a(8) = binomial(10,2)/3 = 5*3 = 15 (example for n == 2 (mod 3)).
		

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, A107711, A026741 (third column of A107711), A109007 (gcd(n,3)).

Programs

Mathematica

Table[Binomial[n + 2, 2] GCD[n + 3, 3]/3, {n, 0, 60}] (* Bruno Berselli, Feb 24 2014 *)
CoefficientList[Series[(1 + x + 2 x^2 + 7 x^3 + 2 x^4 + x^5 + x^6)/(1 - x^3)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 26 2014 *)

PARI

a(n) = numerator((n+1)*(n+2)/6); \\ Altug Alkan, Apr 19 2018

Formula

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

a(n) = A107711(n+3,3) for n >= 0.

a(n) = (2+(-1)^(n+floor((n+1)/3)))*(n+1)*(n+2)/6. - Bruno Berselli, Feb 24 2014

a(n) is the numerator of (n+1)*(n+2)/6. - Altug Alkan, Apr 19 2018

Sum_{n>=0} 1/a(n) = 6 - 4*Pi/(3*sqrt(3)). - Amiram Eldar, Aug 11 2022