A335624 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3*y + 4*z a square, where x, y, z, w are nonnegative integers.

1, 3, 3, 1, 1, 3, 3, 1, 0, 3, 4, 2, 2, 3, 2, 2, 4, 6, 3, 1, 4, 3, 2, 1, 0, 6, 8, 5, 2, 4, 7, 3, 3, 4, 7, 5, 3, 6, 6, 2, 0, 9, 7, 1, 1, 4, 6, 2, 2, 4, 9, 7, 4, 5, 7, 7, 2, 6, 6, 4, 3, 7, 11, 3, 1, 8, 15, 5, 5, 6, 6, 4, 2, 6, 6, 9, 5, 7, 5, 4, 4, 12, 12, 7, 6, 6, 8, 7, 2, 6, 12, 3, 6, 8, 8, 3, 3, 9, 10, 7, 6

Offset

0,2

Comments

Conjecture: a(n) > 0 if n is not divisible by 8. Moreover, a(n) = 0 if and only if n has the form 2^(4k+3)*m (k >= 0 and m = 1, 3, 5, 43).

We have verified this for n up to 3*10^6. The conjecture is similar to the author's 1-3-5 conjecture (cf. A271518).

In his 2017 JNT paper, the author conjectured that any natural number not of the form 2^(4k+2)*7 (k = 0,1,...) can be written as w^2 + x^2 + y^2 + z^2 with w + 2*x + 3*y + 5*z a square, where w, x, y, z are nonnegative integers.

See also A338019 for a similar conjecture.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].

Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

Maple

a(4) = 1, and 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 + 3*0 + 4*0 = 0^2.
a(7) = 1, and 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2 + 3*1 + 4*1 = 3^2.
a(44) = 1, and 44 = 3^2 + 3^2 + 1^2 + 5^2 with 3 + 3*3 + 4*1 = 4^2.
a(328) = 1, and 328 = 8^2 + 16^2 + 2^2 + 2^2 with 8 + 3*16 + 4*2 = 8^2.
a(776) = 1, and 776 = 24^2 + 0^2 + 10^2 + 10^2 with 24 + 3*0 + 4*10 = 8^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+3y+4z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[n-x^2-y^2]}];
tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000118, A000290, A271518, A338019.

Sequence in context: A124039 A350617 A337743 * A096433 A084101 A053386

Adjacent sequences: A335621 A335622 A335623 * A335625 A335626 A335627

Keyword

nonn

Author

Zhi-Wei Sun, Oct 08 2020

Status

approved