A297788 Number of partitions of n into 3 squares and a nonnegative cube.

1, 2, 2, 2, 2, 2, 2, 1, 2, 4, 4, 3, 3, 3, 3, 1, 2, 5, 5, 4, 3, 3, 3, 1, 2, 5, 6, 6, 4, 4, 5, 2, 3, 6, 6, 6, 5, 6, 5, 3, 3, 7, 6, 4, 6, 6, 6, 2, 3, 7, 6, 7, 6, 7, 8, 3, 4, 6, 6, 6, 5, 6, 8, 4, 4, 9, 8, 8, 7, 8, 7, 2, 6, 10, 9, 8, 8, 9, 7, 2, 6, 12, 11, 8, 7, 7

Offset: 0

Author

XU Pingya, Jan 06 2018

Keywords

• nonn

Comments

When n is not of the form 4^a * (8b + 7), according to Legendre's three-square theorem, n = x^2 + y^2 + z^2 = x^2 + y^2 + z^2 + 0^3 (where a, b, x, y and z are nonnegative integers with x <= y <= z).

If n = 8b + 7, then n - 1 = 8b + 6 is not of the form 4^a * (8b + 7). So n = (n - 1) + 1 = x^2 + y^2 + z^2 + 1^3.

If n = 4 * (8b + 7), then n - 1 = 8 * (4b + 3) + 3 is also not of the form 4^a * (8b + 7).

If n = 4^2 * (8b + 7), then n - 8 = 4 * (8 * (4b + 3) + 2) is not of the form 4^a * (8b + 7). n = (n - 8) + 8 = x^2 + y^2 + z^2 + 2^3.

If n = 4^k * (8b + 7) (k >= 3), then n - 8 = 4 * (8 * (4^(k - 1) * b + 4^(k - 3) * 14) - 2) = 4 * (8m - 2) is also not of the form 4^a * (8b + 7).

That is, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative cube, so a(n) > 0.

Examples

			2 = 0^2 + 0^2 + 1^2 + 1^3 = 0^2 + 1^2 + 0^2 + 1^3, a(2) = 2.
9 = 0^2 + 0^2 + 1^2 + 2^3 = 0^2 + 1^2 + 0^2 + 2^3 = 0^2 + 2^2 + 2^2 + 1^3 = 1^2 + 2^2 + 2^2 + 0^3, a(9) = 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Wikipedia, Legendre's three-square theorem

Crossrefs

Cf. A002635, A004215, A274274.

Programs

Maple

N:= 100: # to get a(0)..a(N)
A:= Array(0..N):
for x from 0 to floor(sqrt(N)) do
  for y from 0 to x while x^2 + y^2 <= N do
    for z from 0 to y while x^2 + y^2 + z^2 <= N do
      for w from 0 do
        t:= x^2 + y^2 + z^2 + w^3;
        if t > N then break fi;
        A[t]:= A[t]+1;
od od od od:
convert(A,list); # Robert Israel, Jan 11 2018

Mathematica

a[n_]:=Sum[If[x^2+y^2+z^2+w^3==n, 1, 0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/3)}]
Table[a[n], {n,0,86}]