A117066 Partial sums of cupolar numbers (1/3)*(n+1)*(5*n^2+7*n+3) (A096000).

1, 11, 48, 140, 325, 651, 1176, 1968, 3105, 4675, 6776, 9516, 13013, 17395, 22800, 29376, 37281, 46683, 57760, 70700, 85701, 102971, 122728, 145200, 170625, 199251, 231336, 267148, 306965, 351075, 399776, 453376, 512193, 576555, 646800, 723276, 806341

Offset: 1

Author

Jonathan Vos Post, Apr 17 2006

Keywords

• nonn
• easy

Comments

Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and q. - Wesley Ivan Hurt, Apr 15 2018

Examples

			For n=6, 9*a(6) = 6^3 + 7^3 + 8^3 + 9^3 + 10^3 + 11^3 +12^3 = 9*651. - _Bruno Berselli_, Apr 01 2014
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A005902, A096000.

Programs

GAP

List([1..40], n-> n^2*(n+1)*(5*n+1)/12); # G. C. Greubel, Jul 05 2019

Magma

[n^2*(n+1)*(5*n+1)/12: n in [1..40]]; // Vincenzo Librandi, Apr 16 2018

Maple

a:=n->sum ((n+j)^3, j=0..n): seq(a(n)/9, n=1..40);# Zerinvary Lajos, Dec 17 2008

Mathematica

Table[Sum[n i (n + i), {i, 0, n}]/2, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
Accumulate[Table[((n+1)(5n^2+7n+3))/3,{n,0,50}]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,11,48,140,325},50] (* Harvey P. Dale, Jan 03 2024 *)

PARI

my(x='x+O('x^40)); Vec(x*(3*x^2+6*x+1)/(1-x)^5) \\ G. C. Greubel, Jul 19 2017

PARI

a(n) = n^2*(n+1)*(5*n+1)/12; \\ Altug Alkan, Apr 16 2018

Sage

[n^2*(n+1)*(5*n+1)/12 for n in (1..40)] # G. C. Greubel, Jul 05 2019

Formula

a(n) = Sum_{i=1..n} A096000(i).

a(n) = Sum_{i=1..n} (1/3)*(i+1)*(5*i^2+7*i+3).

a(n) = Sum_{i=1..n} (1/2)*(Q(i) + 3*i^2 + 3*i + 1), where Q(i) are the cuboctahedral numbers (A005902).

a(n) = Sum_{i=0..n} A073254(n,i)*i. - Peter Luschny, Oct 29 2011

G.f.: x*(1+6*x+3*x^2) / (1-x)^5. - Colin Barker, May 08 2013

9*a(n) = Sum_{i=0..n} (n+i)^3, see Maple code by Zerinvary Lajos. - Bruno Berselli, Apr 01 2014

a(n) = n^2*(n+1)*(5*n+1)/12. - Vaclav Kotesovec, Jan 03 2017

E.g.f.: (x/12)*(12 + 54*x + 36*x^2 + 5*x^3)*exp(x). - G. C. Greubel, Jul 19 2017

Another identity: ..., a(4) = (1/2)*(7*(2+4+6+8)+5*(4+6+8)+3*(6+8)+1*(8)) = 140, a(5) = (1/2)*(9*(2+4+6+8+10)+7*(4+6+8+10)+5*(6+8+10)+3*(8+10)+1*(10)) = 325, ... - J. M. Bergot, Aug 25 2022

Extensions

Terms corrected by Colin Barker, May 08 2013