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A063725 Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n. 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Index entries for sequences related to sums of squares

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

EXAMPLE

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

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 *)

PROG

(Haskell)

a063725 n =

   sum $ map (a010052 . (n -)) $ takeWhile (< n) $ tail a000290_list

a063725_list = map a063725 [0..]

-- Reinhard Zumkeller, Aug 16 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

CROSSREFS

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

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

Sequence in context: A321912 A092303 A329343 * A084888 A091400 A129448

Adjacent sequences:  A063722 A063723 A063724 * A063726 A063727 A063728

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 23 2001

STATUS

approved

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Last modified November 18 22:36 EST 2019. Contains 329305 sequences. (Running on oeis4.)