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A033455 Convolution of nonzero squares A000290 with themselves.

%I A033455 #87 Aug 22 2025 20:03:12
%S A033455 1,8,34,104,259,560,1092,1968,3333,5368,8294,12376,17927,25312,34952,
%T A033455 47328,62985,82536,106666,136136,171787,214544,265420,325520,396045,
%U A033455 478296,573678,683704,809999,954304,1118480,1304512,1514513,1750728,2015538,2311464
%N A033455 Convolution of nonzero squares A000290 with themselves.
%C A033455 Total area of all square regions from an n X n grid. E.g., at n = 3, there are nine individual squares, four 2 X 2's and one 3 X 3, total area 9 + 16 + 9 = 34, hence a(3) = 34. - _Jon Perry_, Jul 29 2003
%C A033455 If X is an n-set and Y and Z disjoint 2-subsets of X then a(n) is equal to the number of 7-subsets of X intersecting both Y and Z. - _Milan Janjic_, Aug 26 2007
%C A033455 Every fourth term is odd. However, there are no primes in the sequence. - _Zak Seidov_, Feb 28 2011
%C A033455 -120*a(n) is the real part of (n + n*i)*(n + 2 + n*i)*(n + (n + 2)i)*(n + 2+(n + 2)*i)*(n + 1 + (n + 1)*i), where i = sqrt(-1). - _Jon Perry_, Feb 05 2014
%C A033455 The previous formula rephrases the factorization of the 5th-order polynomial a(n) = (n+1)*((n+1)^4-1) = (n+1)*A123864(n+1) based on the factorization in A123865. - _R. J. Mathar_, Feb 08 2014
%H A033455 Vincenzo Librandi, <a href="/A033455/b033455.txt">Table of n, a(n) for n = 1..1000</a>
%H A033455 Abderrahim Arabi, Hacène Belbachir, Jean-Philippe Dubernard, <a href="https://arxiv.org/abs/1811.05707">Plateau Polycubes and Lateral Area</a>, arXiv:1811.05707 [math.CO], 2018. See Column 1 Table 1 p. 8.
%H A033455 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A033455 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A033455 C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
%H A033455 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A033455 a(n-1) = n*(n^4 - 1)/30 = A061167(n)/30. - _Henry Bottomley_, Apr 18 2001
%F A033455 G.f.: x*(1+x)^2/(1-x)^6. - _Philippe Deléham_, Feb 21 2012
%F A033455 a(n) = Sum_{k=1..n+1} k^2*(n+1-k)^2. - _Kolosov Petro_, Feb 07 2019
%F A033455 E.g.f.: x*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x)/30. - _G. C. Greubel_, Jul 05 2019
%t A033455 Table[(n^5 - n)/30, {n,2,41}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 28 2011 *)
%t A033455 CoefficientList[Series[(1+x)^2/(1-x)^6, {x,0,40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%o A033455 (Magma) [(n^5-n)/30: n in [2..41]]; // _Vincenzo Librandi_, Mar 24 2014
%o A033455 (PARI) vector(40, n, ((n+1)^5 -n-1)/30) \\ _G. C. Greubel_, Jul 05 2019
%o A033455 (Sage) [((n+1)^5 -n-1)/30 for n in (1..40)] # _G. C. Greubel_, Jul 05 2019
%o A033455 (GAP) List([1..40], n-> ((n+1)^5 -(n+1))/30); # _G. C. Greubel_, Jul 05 2019
%Y A033455 Cf. A000290, A001249, A123865, A219086.
%K A033455 nonn,easy,changed
%O A033455 1,2
%A A033455 _N. J. A. Sloane_
%E A033455 More terms from _Vincenzo Librandi_, Mar 24 2014