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

0,29

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

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

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 *)

Crossrefs

Cf. A002476, A092572, A092575.

Sequence in context: A109898 A223074 A192732 * A307801 A186717 A344833

Adjacent sequences: A092570 A092571 A092572 * A092574 A092575 A092576

Keyword

nonn

Author

Eric W. Weisstein, Feb 28 2004

Extensions

Definition corrected by David A. Corneth, Apr 03 2017

Status

approved