A092573 Number of solutions to x^2 + 3y^2 = n in positive integers x and y.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0

Offset: 0

Author

Eric W. Weisstein, Feb 28 2004

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
• Eric Weisstein's World of Mathematics, Euler's 6n+1 Theorem

Crossrefs

Cf. A002476, A092572, A092575.

Programs

Maple

N:= 300: # to get a(0)..a(N)
V:= Vector(N):
for y from 1 to floor(sqrt(N/3-1)) do
  js:= [seq(x^2+3*y^2, x=1..floor(sqrt(N-3*y^2)))];
  V[js]:= map(`+`,V[js],1);
od:
0,op(convert(V,list)); # Robert Israel, Apr 03 2017

Mathematica

r[z_] := Reduce[x > 0 && y > 0 && x^2 + 3 y^2 == z, {x, y}, Integers]; Table[rz = r[z]; If[rz === False, 0, If[rz[[0]] === Or, Length[rz], 1]], {z, 0, 102}] (* Jean-François Alcover, Oct 23 2012 *)
gf = (EllipticTheta[3, 0, x]-1)*(EllipticTheta[3, 0, x^3]-1)/4 + O[x]^105;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2018, after Robert Israel *)

Formula

a(n) = ( A033716(n) - A000122(n) - A000122(n/3) + A000007(n) )/4. - Max Alekseyev, Sep 29 2012

G.f.: (Theta_3(0,x)-1)*(Theta_3(0,x^3)-1)/4 where Theta_3 is a Jacobi theta function. - Robert Israel, Apr 03 2017

Extensions

Definition corrected by David A. Corneth, Apr 03 2017