A303383 Total volume of all cubes with side length q such that n = p + q and p <= q.

0, 1, 8, 35, 91, 216, 405, 748, 1196, 1925, 2800, 4131, 5643, 7840, 10241, 13616, 17200, 22113, 27216, 34075, 41075, 50336, 59653, 71820, 83916, 99541, 114920, 134603, 153811, 178200, 201825, 231616, 260288, 296225, 330616, 373491, 414315, 464968, 512981

Offset

1,3

Links

Table of n, a(n) for n=1..39.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Formula

a(n) = Sum_{i=1..floor(n/2)} (n-i)^3.

From Bruno Berselli, Apr 23 2018: (Start)

G.f.: x*(1 + x + x^2)*(1 + 6*x + 16*x^2 + 6*x^3 + x^4)/((1 - x)^5*(1 + x)^4).

a(n) = (30*(n - 2)*(n + 1)*(n^2 - n + 2) + (2*n - 1)*(2*n^2 - 2*n - 1)*(-1)^n + 119)/128. Therefore:

a(n) = n^2*(3*n - 2)*(5*n - 6)/64 for n even;

a(n) = (n - 1)^2*(3*n - 1)*(5*n + 1)/64 for n odd. (End)

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9). - Wesley Ivan Hurt, Apr 23 2021

Mathematica

Table[Sum[(n - i)^3, {i, Floor[n/2]}], {n, 50}]

Prog

(Magma) [0] cat [&+[(n-k)^3: k in [1..n div 2]]: n in [2..80]]; // Vincenzo Librandi, Apr 23 2018

Crossrefs

Cf. A105636.

Subsequence of A217843.

After 8, all terms belong to A265377.

Sequence in context: A257454 A305672 A100907 * A265840 A212903 A168566

Adjacent sequences: A303380 A303381 A303382 * A303384 A303385 A303386

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 22 2018

Status

approved