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A025427 Number of partitions of n into 3 nonzero squares. 37

%I

%S 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,

%T 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,

%U 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

%N Number of partitions of n into 3 nonzero squares.

%C 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

%H R. J. Mathar and R. Zumkeller, <a href="/A025427/b025427.txt">Table of n, a(n) for n = 0..10000</a>, first 5592 terms from R. J. Mathar

%H <a href="/index/Su#ssq">Index to sequences related to sums of squares and cubes</a>.

%F 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

%F 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

%F 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

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

%e 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).

%p A025427 := proc(n)

%p local a,x,y,zsq ;

%p a := 0 ;

%p for x from 1 do

%p if 3*x^2 > n then

%p return a;

%p end if;

%p for y from x do

%p if x+2*y^2 > n then

%p break;

%p end if;

%p zsq := n-x^2-y^2 ;

%p if issqr(zsq) then

%p a := a+1 ;

%p end if;

%p end do:

%p end do:

%p end proc: # _R. J. Mathar_, Sep 15 2015

%t Count[PowersRepresentations[#, 3, 2], pr_ /; (Times @@ pr) > 0]& /@ Range[0, 120] (* _Jean-Fran├žois Alcover_, Jan 30 2018 *)

%o (Haskell)

%o a025427 n = sum $ map f zs where

%o f x = sum $ map (a010052 . (n - x -)) $

%o takeWhile (<= div (n - x) 2) $ dropWhile (< x) zs

%o zs = takeWhile (< n) $ tail a000290_list

%o -- _Reinhard Zumkeller_, Feb 26 2015

%Y Cf. A000408, A024795, A223730 (multiplicities for the primitive case). - _Wolfdieter Lang_, Apr 03 2013

%Y Column k=3 of A243148.

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

%K nonn,easy

%O 0,28

%A _David W. Wilson_

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Last modified June 18 17:05 EDT 2019. Contains 324214 sequences. (Running on oeis4.)