A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

1, 2, 3, 2, 2, 2, 2, 1, 2, 4, 5, 3, 2, 3, 2, 1, 3, 5, 6, 3, 3, 3, 2, 0, 2, 5, 6, 5, 4, 5, 2, 2, 4, 5, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 7, 3, 2, 4, 3, 5, 4, 7, 8, 5, 3, 3, 3, 5, 5, 5, 6, 4, 3, 6, 7, 8, 7, 5, 7, 4, 2, 7, 9, 10, 4, 5, 7, 3, 3, 9, 10, 8, 5, 4, 7

Offset: 0

Author

XU Pingya, Jan 08 2018

Keywords

• nonn

Comments

For n <= 6 * 10^7, except for a(23) = 0, all a(n) > 0.

First occurrence of k beginning with 0: 23, 7, 1, 2, 9, 10, 18, 45, 53, 73, 74, 101, 125, 146, 165, 197, ..., . - Robert G. Wilson v, Jan 14 2018

Examples

			2 = 0^2 + 0^2 + 1^3 + 1^3 = 0^2 + 1^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 0^3, a(2) = 3.
10 = 0^2 + 1^2 + 1^3 + 2^3 = 0^2 + 3^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 2^3 = 1^2 + 3^2 + 0^3 + 0^3 = 2^2 + 2^2 + 1^3 + 1^3, a(10) = 5.
		

Links

• W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.
• Eric Weisstein's World of Mathematics, Waring's Problem

Crossrefs

Cf. A000164, A002635, A022552, A274274.

Programs

Mathematica

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