A025435 Number of partitions of n into 2 distinct squares.

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

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(A004435(n)) = 0; a(A001983(n)) > 0. - Reinhard Zumkeller, Dec 20 2013

Examples

			G.f. = x + x^4 + x^5 + x^9 + x^10 + x^13 + x^16 + x^17 + x^20 + 2*x^25 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A010052, A000290, A000161, A025441.

Programs

Haskell

a025435 0 = 0
a025435 n = a010052 n + sum
   (map (a010052 . (n -)) $ takeWhile (< n `div` 2) $ tail a000290_list)
-- Reinhard Zumkeller, Dec 20 2013

Maple

A025435 := proc(n)
    local i, j, ans;
    ans := 0;
    for i from 0 to n do
        for j from i+1 to n do
            if i^2+j^2=n then
                ans := ans+1
            fi
        end do
    end do;
    ans ;
end proc: # R. J. Mathar, Aug 04 2018

Mathematica

a[ n_] := If[ n < 0, 0, Sum[ Boole[ n == i^2 + j^2], {i, Sqrt[n]}, {j, 0, i - 1}]]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := Length@ PowersRepresentations[ n, 2, 2] - Boole @ IntegerQ @ Sqrt[2 n]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := SeriesCoefficient[ With[ {f = (EllipticTheta[ 3, 0, x] + 1)/2, g = (EllipticTheta[ 3, 0, x^2] + 1)/2}, f f - g] / 2, {x, 0, n}]; (* Michael Somos, Jun 24 2015 *)

PARI

{a(n) = if( n<0, 0, sum(i=1, sqrtint(n), sum(j=0, i-1, n == i^2 + j^2)))}; /* Michael Somos, Jun 24 2015 */

PARI

A025435(n)=sum(k=sqrtint((n-1+!n)\2)+1, sqrtint(n), issquare(n-k^2))-issquare(n/2) \\ or A000161(n)-issquare(n/2). - M. F. Hasler, Aug 05 2018

Python

from math import prod
from sympy import factorint
def A025435(n):
    f = factorint(n)
    return int(not any(e&1 for p,e in f.items() if p>2))*(1-((f.get(2,0)&1)<<1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # Chai Wah Wu, Sep 08 2022

Formula

a(n) = A000161(n) - A010052(2*n). - M. F. Hasler, Aug 05 2018

a(n) = Sum_{i=1..n} c(i) * c(2*n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020