A051662 House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.

1, 9, 32, 78, 155, 271, 434, 652, 933, 1285, 1716, 2234, 2847, 3563, 4390, 5336, 6409, 7617, 8968, 10470, 12131, 13959, 15962, 18148, 20525, 23101, 25884, 28882, 32103, 35555, 39246, 43184, 47377, 51833, 56560, 61566, 66859, 72447, 78338, 84540, 91061, 97909

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Keywords

• easy
• nice
• nonn

Comments

Binomial transform of [1, 8, 15, 8, 0, 0, 0, ...]. - Gary W. Adamson, Nov 23 2007

Principal diagonal of the convolution array A213751. - Clark Kimberling, Jun 20 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Josh Deprez, Fair amenability for semigroups, arXiv preprint arXiv:1310.5589 [math.GR], 2013-2015.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000330, A220084 (for a list of numbers of the form n*P(k,n) - (n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).

Cf. A000330, A000578, A007318, A023855, A213751.

Programs

Haskell

- following Gary W. Adamson's comment.
a051662 = sum . zipWith (*) [1, 8, 15, 8] . a007318_row
-- Reinhard Zumkeller, Feb 19 2015

Maple

a:=n->sum(k^2, k=1..n):seq(a(n)+sum(n^2, k=2..n), n=1...40); # Zerinvary Lajos, Jun 11 2008

Mathematica

Table[(n+1)^3+Sum[i^2,{i,n}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1}, {1,9,32,78},40] (* Harvey P. Dale, Jun 23 2011 *)

Maxima

A051662(n):=((8*n+21)*n+19)*n/6+1$ makelist(A051662(n),n,0,15); /* Martin Ettl, Dec 13 2012 */

PARI

a(n)=((8*n+21)*n+19)*n/6+1 \\ Charles R Greathouse IV, Jun 23 2011

Formula

a(n) = (n+1)*(8*n^2 + 13*n + 6)/6.

a(n) = A000578(n+1) + A000330(n).

From Harvey P. Dale, Jun 23 2011: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=1, a(1)=9, a(2)=32, a(3)=78.

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

a(n) = (n+1)*A000330(n+1) - n*A000330(n). - Bruno Berselli, Dec 11 2012

a(n) = A023855(2*n) + A023855(2*n+1). - Luc Rousseau, Feb 24 2018

E.g.f.: exp(x)*(6 + 48*x + 45*x^2 + 8*x^3)/6. - Elmo R. Oliveira, Aug 06 2025

Extensions

Corrected by T. D. Noe, Nov 01 2006 and Nov 08 2006