A063725 Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n.

0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0

Offset: 0

Author

N. J. A. Sloane, Aug 23 2001

Keywords

• nonn

Comments

a(A018825(n))=0; a(A000404(n))>0; a(A081324(n))=1; a(A004431(n))>1. - Reinhard Zumkeller, Aug 16 2011

Examples

			a(5) = 2 from the solutions (1,2) and (2,1).
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000404 (the numbers n that can be represented in this form).

Cf. A000161, A063691, A063730, A025426, A000290, A010052.

Column k=2 of A337165.

Programs

Haskell

a063725 n =
   sum $ map (a010052 . (n -)) $ takeWhile (< n) $ tail a000290_list
a063725_list = map a063725 [0..]
-- Reinhard Zumkeller, Aug 16 2011

Mathematica

nn = 100; t = Table[0, {nn}]; s = Sqrt[nn]; Do[n = x^2 + y^2; If[n <= nn, t[[n]]++], {x, s}, {y, s}]; Join[{0}, t] (* T. D. Noe, Apr 03 2011 *)

PARI

a(n)=if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ Charles R Greathouse IV, May 18 2016

Python

from math import prod
from sympy import factorint
def A063725(n):
    f = factorint(n)
    return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items())-(not any(e&1 for e in f.values())) if n else 0 # Chai Wah Wu, May 17 2023

Formula

G.f.: (Sum_{m=1..inf} x^(m^2))^2.

a(n) = ( A004018(n) - 2*A000122(n) + A000007(n) )/4. - Max Alekseyev, Sep 29 2012

G.f.: (theta_3(q) - 1)^2/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018