A303383 - OEIS

%I #22 Sep 08 2022 08:46:21

%S 0,1,8,35,91,216,405,748,1196,1925,2800,4131,5643,7840,10241,13616,

%T 17200,22113,27216,34075,41075,50336,59653,71820,83916,99541,114920,

%U 134603,153811,178200,201825,231616,260288,296225,330616,373491,414315,464968,512981

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).

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

%F From _Bruno Berselli_, Apr 23 2018: (Start)

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

%F 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:

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

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

%F 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

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

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

%Y Cf. A105636.

%Y Subsequence of A217843.

%Y After 8, all terms belong to A265377.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Apr 22 2018