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

%I A051662 #52 Aug 07 2025 04:00:57
%S A051662 1,9,32,78,155,271,434,652,933,1285,1716,2234,2847,3563,4390,5336,
%T A051662 6409,7617,8968,10470,12131,13959,15962,18148,20525,23101,25884,28882,
%U A051662 32103,35555,39246,43184,47377,51833,56560,61566,66859,72447,78338,84540,91061,97909
%N A051662 House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.
%C A051662 Binomial transform of [1, 8, 15, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 23 2007
%C A051662 Principal diagonal of the convolution array A213751. - _Clark Kimberling_, Jun 20 2012
%H A051662 Reinhard Zumkeller, <a href="/A051662/b051662.txt">Table of n, a(n) for n = 0..1000</a>
%H A051662 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 A051662 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A051662 a(n) = (n+1)*(8*n^2 + 13*n + 6)/6.
%F A051662 a(n) = A000578(n+1) + A000330(n).
%F A051662 From _Harvey P. Dale_, Jun 23 2011: (Start)
%F A051662 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.
%F A051662 G.f.: (1+5*x+2*x^2)/(x-1)^4. (End)
%F A051662 a(n) = (n+1)*A000330(n+1) - n*A000330(n). - _Bruno Berselli_, Dec 11 2012
%F A051662 a(n) = A023855(2*n) + A023855(2*n+1). - _Luc Rousseau_, Feb 24 2018
%F A051662 E.g.f.: exp(x)*(6 + 48*x + 45*x^2 + 8*x^3)/6. - _Elmo R. Oliveira_, Aug 06 2025
%p A051662 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 A051662 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 A051662 (PARI) a(n)=((8*n+21)*n+19)*n/6+1 \\ _Charles R Greathouse IV_, Jun 23 2011
%o A051662 (Maxima) A051662(n):=((8*n+21)*n+19)*n/6+1$ makelist(A051662(n),n,0,15); /* _Martin Ettl_, Dec 13 2012 */
%o A051662 (Haskell)  - following Gary W. Adamson's comment.
%o A051662 a051662 = sum . zipWith (*) [1, 8, 15, 8] . a007318_row
%o A051662 -- _Reinhard Zumkeller_, Feb 19 2015
%Y A051662 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 A051662 Cf. A000330, A000578, A007318, A023855, A213751.
%K A051662 easy,nice,nonn
%O A051662 0,2
%A A051662 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%E A051662 Corrected by _T. D. Noe_, Nov 01 2006 and Nov 08 2006