A175368 Number of integer 4-tuples (x,y,z,u) satisfying |x|^3 + |y|^3 + |z|^3 + |u|^3 = n, -n <= x,y,z,u <= n.

1, 8, 24, 32, 16, 0, 0, 0, 8, 48, 96, 64, 0, 0, 0, 0, 24, 96, 96, 0, 0, 0, 0, 0, 32, 64, 0, 8, 48, 96, 64, 0, 16, 0, 0, 48, 192, 192, 0, 0, 0, 0, 0, 96, 192, 0, 0, 0, 0, 0, 0, 64, 0, 0, 24, 96, 96, 0, 0, 0, 0, 0, 96, 192, 8, 48, 96, 64, 0, 0, 96, 0, 48, 192, 192, 0, 0, 0, 0, 0, 96, 224, 64, 0

Offset: 0

Author

R. J. Mathar, Apr 24 2010

Keywords

• nonn

Comments

A variant of A000118 with cubes instead of squares.

Examples

			a(1) = 8 counts (x,y,z,u) = (-1,0,0,0), (0,-1,0,0), (0,0,-1,0), (0,0,0,-1) and 4 more tuples with -1 replaced by +1.
a(2) = 24 counts (x,y,z,u) = (-1,-1,0,0), (-1,0,-1,0), (-1,0,0,-1), (-1,0,0,1) etc, all variants where two of the 4 values are zero and the other two +1 or -1.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000118 , A175362, A175365.

Programs

PARI

a(n, k=4) = if(n==0, return(1)); if(k <= 0, return(0)); if(k == 1, return(ispower(n,3))); my(count = 0); for(v = 0, sqrtnint(n, 3), count += (2 - (v == 0))*if(k > 2, a(n - v^3, k-1), if(ispower(n - v^3, 3), 2 - (n - v^3 == 0), 0))); count; \\ Daniel Suteu, Aug 15 2021

Formula

Conjectured g.f.: (1 + 2*Sum_{j>=1} x^(j^3))^4.

a(n) = A175365(n) + 2*Sum_{k=1..floor(n^(1/3))} A175365(n - k^3). - Daniel Suteu, Aug 15 2021