A101357 Partial sums of A060354.

0, 1, 3, 9, 25, 60, 126, 238, 414, 675, 1045, 1551, 2223, 3094, 4200, 5580, 7276, 9333, 11799, 14725, 18165, 22176, 26818, 32154, 38250, 45175, 53001, 61803, 71659, 82650, 94860, 108376, 123288, 139689, 157675, 177345, 198801, 222148, 247494

Offset: 0

Author

Jonathan Vos Post, Dec 25 2004

Keywords

• easy
• nonn

Comments

The Ca4 triangle sums of A139600 are given by the terms of this sequence. For the definitions of the Ca4 and other triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Polygonal number
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A060354, A000332.

Programs

Magma

[(n^4-2*n^3+3*n^2+6*n)/8: n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Mathematica

Table[Sum[(i*(i - 2)^2 + i^2)/2, {i, 0, n}], {n, 0, 38}]
Accumulate[Table[(n (n-2)^2+n^2)/2,{n,0,50}]] (* Harvey P. Dale, Aug 05 2011 *)

PARI

a(n)=(n^4-2*n^3+3*n^2+6*n)/8 \\ Charles R Greathouse IV, Oct 16 2015

Formula

a(n) = Sum_{i=0..n} (i(i-2)^2 + i^2)/2.

a(n) = A004255(n), n > 0. - R. J. Mathar, Sep 02 2008

a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 4*binomial(n+1,4).

a(n) = (n^4 - 2*n^3 + 3*n^2 + 6*n)/8. - Johannes W. Meijer, Apr 29 2011

G.f.: -x*(4*x^2 - 2*x + 1) / (x-1)^5. - Colin Barker, Apr 29 2013

Extensions

More terms from Joshua Zucker, May 12 2006

Edited by Stefan Steinerberger, Aug 01 2007