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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

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).

Crossrefs

Cf. A000290, A001249, A123865, A219086.

Programs

GAP

List([1..40], n-> ((n+1)^5 -(n+1))/30); # G. C. Greubel, Jul 05 2019

Magma

[(n^5-n)/30: n in [2..41]]; // Vincenzo Librandi, Mar 24 2014

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 *)

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

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

Extensions

More terms from Vincenzo Librandi, Mar 24 2014