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

0,2

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.

Links

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

Wikipedia, Legendre's three-square theorem

Example

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.

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}]

Crossrefs

Cf. A002635, A004215, A274274.

Sequence in context: A194318 A378123 A306459 * A194342 A230410 A044925

Adjacent sequences: A297785 A297786 A297787 * A297789 A297790 A297791

Keyword

nonn

Author

XU Pingya, Jan 06 2018

Status

approved