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

%I A366091 #11 Sep 29 2023 16:13:17
%S A366091 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,
%T A366091 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,
%U A366091 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
%N A366091 a(n) is the number of ways to write n = i^2 + 2*j^2 + 3*k^2 with i,j,k >= 0.
%H A366091 Robert Israel, <a href="/A366091/b366091.txt">Table of n, a(n) for n = 0..10000</a>
%F A366091 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.
%e A366091 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.
%p A366091 g:= (1+JacobiTheta3(0,z))*(1+JacobiTheta3(0,z^2))*(1+JacobiTheta3(0,z^3))/8:
%p A366091 S:= series(g,z,101):
%p A366091 seq(coeff(S,z,j),j=0..100);
%o A366091 (Python)
%o A366091 from itertools import count
%o A366091 from sympy.ntheory.primetest import is_square
%o A366091 def A366091(n):
%o A366091     c = 0
%o A366091     for k in count(0):
%o A366091         if (a:=3*k**2)>n:
%o A366091             break
%o A366091         for j in count(0):
%o A366091             if (b:=a+(j**2<<1))>n:
%o A366091                 break
%o A366091             if is_square(n-b):
%o A366091                 c += 1
%o A366091     return c # _Chai Wah Wu_, Sep 29 2023
%Y A366091 Cf.  A028594 (allows any integer i,j,k), A055042 (a(n) = 0)
%K A366091 nonn
%O A366091 0,4
%A A366091 _Robert Israel_, Sep 28 2023