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URL: https://oeis.org/A051662

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A051662
House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.
9
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
0,2
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
Josh Deprez, Fair amenability for semigroups, arXiv preprint arXiv:1310.5589 [math.GR], 2013-2015.
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
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 *)
PROG
(PARI) a(n)=((8*n+21)*n+19)*n/6+1 \\ Charles R Greathouse IV, 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 */
(Haskell) - following Gary W. Adamson's comment.
a051662 = sum . zipWith (*) [1, 8, 15, 8] . a007318_row
-- Reinhard Zumkeller, Feb 19 2015
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).
Sequence in context: A063134 A027620 A152619 * A326247 A362526 A225918
KEYWORD
easy,nice,nonn
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
EXTENSIONS
Corrected by T. D. Noe, Nov 01 2006 and Nov 08 2006
STATUS
approved