A025427 Number of partitions of n into 3 nonzero squares.

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

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

The non-vanishing values a(n) give the multiplicities for the numbers n appearing in A000408. See also A024795 where these numbers n are listed a(n) times. For the primitive case see A223730 and A223731. - Wolfdieter Lang, Apr 03 2013

Examples

			a(27) = 2 because  1^2 + 1^2 + 5^2 = 27  = 3^2 + 3^2 + 3^2. The second representation is not primitive (gcd(3,3,3) = 3 not 1).
		

Links

• R. J. Mathar and R. Zumkeller, Table of n, a(n) for n = 0..10000, first 5592 terms from R. J. Mathar
• Index to sequences related to sums of squares and cubes.

Crossrefs

Cf. A000408, A024795, A223730 (multiplicities for the primitive case). - Wolfdieter Lang, Apr 03 2013

Column k=3 of A243148.

Cf. A000290, A010052, A004214, A025321, A025414, A025426.

Programs

Haskell

a025427 n = sum $ map f zs where
   f x = sum $ map (a010052 . (n - x -)) $
                   takeWhile (<= div (n - x) 2) $ dropWhile (< x) zs
   zs = takeWhile (< n) $ tail a000290_list
-- Reinhard Zumkeller, Feb 26 2015

Maple

A025427 := proc(n)
    local a,x,y,zsq ;
    a := 0 ;
    for x from 1 do
        if 3*x^2 > n then
            return a;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            end if;
            zsq := n-x^2-y^2 ;
            if issqr(zsq) then
                a := a+1 ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 3):
seq(a(n), n=0..107);  # Alois P. Heinz, Jun 14 2025

Mathematica

Count[PowersRepresentations[#, 3, 2], pr_ /; (Times @@ pr) > 0]& /@ Range[0, 120] (* Jean-François Alcover, Jan 30 2018 *)

PARI

a(n)=if(n<3, return(0)); sum(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); sum(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), issquare(t-j^2))) \\ Charles R Greathouse IV, Aug 05 2024

Formula

a(A004214(n)) = 0; a(A000408(n)) > 0; a(A025414(n)) = n and a(m) != n for m < A025414(n). - Reinhard Zumkeller, Feb 26 2015

a(4n) = a(n). This is because if a number divisible by 4 is the sum of three squares, each of those squares must be even. - Robert Israel, Mar 09 2016

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010052(i) * A010052(k) * A010052(n-i-k). - Wesley Ivan Hurt, Apr 19 2019

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019