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

%I A063725 #34 May 17 2023 10:24:43
%S A063725 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,
%T A063725 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,
%U A063725 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
%N A063725 Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n.
%C A063725 a(A018825(n))=0; a(A000404(n))>0; a(A081324(n))=1; a(A004431(n))>1. - _Reinhard Zumkeller_, Aug 16 2011
%H A063725 T. D. Noe, <a href="/A063725/b063725.txt">Table of n, a(n) for n = 0..10000</a>
%H A063725 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A063725 G.f.: (Sum_{m=1..inf} x^(m^2))^2.
%F A063725 a(n) = ( A004018(n) - 2*A000122(n) + A000007(n) )/4. - _Max Alekseyev_, Sep 29 2012
%F A063725 G.f.: (theta_3(q) - 1)^2/4, where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Aug 08 2018
%e A063725 a(5) = 2 from the solutions (1,2) and (2,1).
%t A063725 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 *)
%o A063725 (Haskell)
%o A063725 a063725 n =
%o A063725    sum $ map (a010052 . (n -)) $ takeWhile (< n) $ tail a000290_list
%o A063725 a063725_list = map a063725 [0..]
%o A063725 -- _Reinhard Zumkeller_, Aug 16 2011
%o A063725 (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
%o A063725 (Python)
%o A063725 from math import prod
%o A063725 from sympy import factorint
%o A063725 def A063725(n):
%o A063725     f = factorint(n)
%o A063725     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
%Y A063725 Cf. A000404 (the numbers n that can be represented in this form).
%Y A063725 Cf. A000161, A063691, A063730, A025426, A000290, A010052.
%Y A063725 Column k=2 of A337165.
%K A063725 nonn
%O A063725 0,6
%A A063725 _N. J. A. Sloane_, Aug 23 2001