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House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.
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%I #48 Mar 18 2018 09:56:47

%S 1,9,32,78,155,271,434,652,933,1285,1716,2234,2847,3563,4390,5336,

%T 6409,7617,8968,10470,12131,13959,15962,18148,20525,23101,25884,28882,

%U 32103,35555,39246,43184,47377,51833,56560,61566,66859,72447,78338,84540

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

%C Binomial transform of [1, 8, 15, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 23 2007

%C Principal diagonal of the convolution array A213751. - _Clark Kimberling_, Jun 20 2012

%H Reinhard Zumkeller, <a href="/A051662/b051662.txt">Table of n, a(n) for n = 0..1000</a>

%H Josh Deprez, <a href="http://arxiv.org/abs/1310.5589">Fair amenability for semigroups</a>, arXiv preprint arXiv:1310.5589 [math.GR], 2013-2015.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F a(0)=1, a(1)=9, a(2)=32, a(3)=78, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Jun 23 2011

%F G.f.: (1+5*x+2*x^2)/(x-1)^4. - _Harvey P. Dale_, Jun 23 2011

%F a(n) = (n+1)*A000330(n+1) - n*A000330(n). - _Bruno Berselli_, Dec 11 2012

%F a(n) = A023855(2*n) + A023855(2*n+1). - _Luc Rousseau_, Feb 24 2018

%p 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

%t 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 *)

%o (PARI) a(n)=((8*n+21)*n+19)*n/6+1 \\ _Charles R Greathouse IV_, Jun 23 2011

%o (Maxima) A051662(n):=((8*n+21)*n+19)*n/6+1$ makelist(A051662(n),n,0,15); /* _Martin Ettl_, Dec 13 2012 */

%o (Haskell) - following Gary W. Adamson's comment.

%o a051662 = sum . zipWith (*) [1, 8, 15, 8] . a007318_row

%o -- _Reinhard Zumkeller_, Feb 19 2015

%Y A000578(n+1)+A000330(n).

%Y 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).

%Y Cf. A007318.

%K easy,nice,nonn

%O 0,2

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

%E Corrected by _T. D. Noe_, Nov 01 2006 and Nov 08 2006