A175362 Number of integer pairs (x,y) satisfying |x|^3 + |y|^3 = n, -n <= x,y <= n.

1, 4, 4, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

R. J. Mathar, Apr 24 2010

Keywords

• nonn

Comments

Cube variant of A004018.

Obviously, a(n) must be 4*k, for k >= 0, n > 0. - Altug Alkan, Apr 09 2016

From Robert Israel, Jan 26 2017: (Start)

a(k^3*n) >= a(n) for k >= 1.

a(n) >= 16 for n in A001235.

a(A011541(n)) >= 8*n. (End)

Examples

			a(2) = 4 counts (x,y) = (-1,1), (1,1), (-1,-1) and (1,-1).
a(9) = 8 counts (x,y) = (-2,-1), (-2,1), (-1,-2), (-1,2), (1,-2), (1,2), (2,-1) and (2,1).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001235, A010057, A011541, A025446, A025455, A025464, A025468, A121980.

Programs

Maple

N:= 200: # to get a(0)..a(N)
G:= (1+2*add(x^(j^3),j=1..floor(N^(1/3))))^2:
seq(coeff(G,x,j),j=0..N); # Robert Israel, Jan 26 2017

PARI

a(n) = if(n==0, 1, 4*sum(k=1, sqrtnint(n, 3), ispower(n - k^3, 3))); \\ Daniel Suteu, Aug 16 2021

Formula

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

a(n^3) = 4 for n > 0. - Altug Alkan, Apr 09 2016

a(n) = 4*Sum_{k=1..floor(n^(1/3))} A010057(n - k^3), for n > 0. - Daniel Suteu, Aug 15 2021

Extensions

Invalid claim that belonged to A004018 removed by R. J. Mathar, Apr 24 2010