A366091 a(n) is the number of ways to write n = i^2 + 2*j^2 + 3*k^2 with i,j,k >= 0.

1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 0, 2, 4, 1, 2, 2, 2, 1, 3, 2, 2, 4, 2, 1, 2, 2, 0, 4, 3, 2, 5, 2, 1, 3, 2, 2, 7, 2, 2, 5, 0, 2, 0, 2, 4, 4, 3, 1, 4, 3, 3, 5, 3, 2, 7, 1, 2, 6, 0, 3, 6, 2, 2, 4, 2, 2, 6, 3, 2, 4, 3, 3, 3, 2, 0, 7, 5, 2, 6, 3, 2, 8, 2, 2, 11, 2, 5, 2, 2, 3, 0, 4, 3, 7, 3, 2, 2, 3, 3

Offset: 0

Author

Robert Israel, Sep 28 2023

Keywords

• nonn

Examples

			a(9) = 3 because 9 = 3^2 + 2*0^2 + 3*0^2 = 1^2 + 2*2^2 + 3*0^2 = 2^2 + 2*1^2 + 3*1^2.
		

Links

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

Crossrefs

Cf. A028594 (allows any integer i,j,k), A055042 (a(n) = 0)

Programs

Maple

g:= (1+JacobiTheta3(0,z))*(1+JacobiTheta3(0,z^2))*(1+JacobiTheta3(0,z^3))/8:
S:= series(g,z,101):
seq(coeff(S,z,j),j=0..100);

Python

from itertools import count
from sympy.ntheory.primetest import is_square
def A366091(n):
    c = 0
    for k in count(0):
        if (a:=3*k**2)>n:
            break
        for j in count(0):
            if (b:=a+(j**2<<1))>n:
                break
            if is_square(n-b):
                c += 1
    return c # Chai Wah Wu, Sep 29 2023

Formula

G.f. (1 + theta_3(0,z)) * (1 + theta_3(0,z^2)) * (1 + theta_3(0,z^3))/8 where theta_3 is a Jacobi theta function.