A033455 Convolution of nonzero squares A000290 with themselves.

1, 8, 34, 104, 259, 560, 1092, 1968, 3333, 5368, 8294, 12376, 17927, 25312, 34952, 47328, 62985, 82536, 106666, 136136, 171787, 214544, 265420, 325520, 396045, 478296, 573678, 683704, 809999, 954304, 1118480, 1304512, 1514513, 1750728, 2015538, 2311464

Offset

1,2

Comments

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

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

Every fourth term is odd. However, there are no primes in the sequence. - Zak Seidov, Feb 28 2011

-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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Abderrahim Arabi, Hacène Belbachir, Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 1 Table 1 p. 8.

Milan Janjic, Two Enumerative Functions

Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - N. J. A. Sloane, Mar 23 2014

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n-1) = n*(n^4 - 1)/30 = A061167(n)/30. - Henry Bottomley, Apr 18 2001

G.f.: x*(1+x)^2/(1-x)^6. - Philippe Deléham, Feb 21 2012

a(n) = Sum_{k=1..n+1} k^2*(n+1-k)^2. - Kolosov Petro, Feb 07 2019

E.g.f.: x*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x)/30. - G. C. Greubel, Jul 05 2019

Mathematica

Table[(n^5 - n)/30, {n, 2, 41}] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2011 *)
CoefficientList[Series[(1+x)^2/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

Prog

(Magma) [(n^5-n)/30: n in [2..41]]; // Vincenzo Librandi, Mar 24 2014
(PARI) vector(40, n, ((n+1)^5 -n-1)/30) \\ G. C. Greubel, Jul 05 2019
(Sage) [((n+1)^5 -n-1)/30 for n in (1..40)] # G. C. Greubel, Jul 05 2019
(GAP) List([1..40], n-> ((n+1)^5 -(n+1))/30) # G. C. Greubel, Jul 05 2019

Crossrefs

Cf. A000290, A001249, A123865, A219086.

Sequence in context: A208639 A240785 A066804 * A172202 A053298 A196311

Adjacent sequences: A033452 A033453 A033454 * A033456 A033457 A033458

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, Mar 24 2014

Status

approved