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

0,26

Comments

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

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A010052, A000290, A000161, A025441.

Sequence in context: A286562 A185644 A319080 * A304685 A186714 A160382

Adjacent sequences: A025432 A025433 A025434 * A025436 A025437 A025438

Keyword

nonn

Author

David W. Wilson

Status

approved