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

0,28

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

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.

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

Example

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).

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*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

Mathematica

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

Prog

(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
(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

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.

Sequence in context: A343653 A066620 A219023 * A348536 A245963 A291375

Adjacent sequences: A025424 A025425 A025426 * A025428 A025429 A025430

Keyword

nonn,easy

Author

David W. Wilson

Status

approved