A081282 Generalized centered polygonal numbers.

0, 1, 6, 22, 62, 147, 308, 588, 1044, 1749, 2794, 4290, 6370, 9191, 12936, 17816, 24072, 31977, 41838, 53998, 68838, 86779, 108284, 133860, 164060, 199485, 240786, 288666, 343882, 407247, 479632, 561968, 655248, 760529, 878934, 1011654

Offset: 0

Author

Paul Barry, Mar 17 2003

Keywords

• nonn
• easy

Comments

Partial sums of A006007.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A081284.

Programs

Magma

[(n*(n+1)*(n+2)*(n^2+2*n+7))/60: n in [0..35]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

CoefficientList[Series[x (1 + x^2) / (1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,6,22,62,147},40] (* Harvey P. Dale, Jan 09 2014 *)

Formula

a(n) = 2*C(n+3, 5) + C(n+2, 3).

G.f.: x*(1+x^2)/(1-x)^6.

a(n) = n*(n+1)*(n+2)*(n^2+2*n+7)/60. - Carl Najafi, Nov 18 2012

a(0)=0, a(1)=1, a(2)=6, a(3)=22, a(4)=62, a(5)=147, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 09 2014