A175375 Number of integer triples (x,y,z) satisfying x^4+y^4+z^4=n, -n <= x,y,z <= n.

1, 6, 12, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 24, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 24, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 24, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 48, 0, 0

Offset: 0

Author

R. J. Mathar, Apr 24 2010

Keywords

• nonn
• look

Comments

4th power variant of A004018 and A175361.

Examples

			a(0) = 1 counts (x,y,z) = (0,0,0). a(3) = 8 counts (x,y,z) = (-1,-1,-1), (-1,-1,1), (-1,1,-1), (-1,1,1), (1,-1,-1), (1,-1,1), (1,1,-1) and (1,1,1). a(17) = 24 counts triples where one of x, y and z is 0, one is +-1 and the third +-2.
		

Links

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

Crossrefs

Cf. A004018, A175361.

Programs

Maple

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
for i from 0 while i^4 <= N do
  if i=0 then ai:= 1 else ai:= 2 fi;
  for j from 0 while i^4 + j^4 <= N do
    if j=0 then aj:= 1 else aj:= 2 fi;
    for k from 0 do
      v:= i^4 + j^4 + k^4;
      if v > N then break fi;
      if k = 0 then ak:= 1 else ak:= 2 fi;
      A[v]:= A[v] + ai*aj*ak;
od od od:
seq(A[i],i=0..N); # Robert Israel, May 01 2019

Formula

G.f.: ( 1 + 2*Sum_{j>0} x^(j^4) )^3.