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A175958 Number of partitions of n^2 into 4 distinct nonzero squares.

%I A175958 #16 Jun 24 2015 06:39:57
%S A175958 0,0,0,0,0,0,0,0,0,1,0,1,0,2,1,5,0,4,4,5,0,10,4,7,0,11,7,17,1,13,17,
%T A175958 15,0,29,13,27,4,23,17,41,0,29,35,32,4,66,24,38,0,47,35,73,7,50,56,73,
%U A175958 1,91,42,63,17,68,49,125,0,103,93,83,13,133,86,93,4
%N A175958 Number of partitions of n^2 into 4 distinct nonzero squares.
%H A175958 Alois P. Heinz and Donovan Johnson, <a href="/A175958/b175958.txt">Table of n, a(n) for n = 0..1000</a> (terms up to a(300) from Alois P. Heinz)
%F A175958 a(n) = A025443(n^2).
%e A175958 a(9)=1 refers to the partition 9^2 = 2^2+4^2+5^2+6^2. a(11)=1 refers to 11^2 = 1^2+2^2+4^2+10^2. a(13)=2 refers to 13^2 = 1^2+2^2+8^2+10^2 = 2^2+4^2+7^2+10^2.
%p A175958 A025443 := proc(n) local res,a,b,c,d ; res := 0 ; for a from 1 do if 4*a^2 > n then break; fi; for b from a+1 do if a^2+3*b^2 > n then break; fi; for c from b+1 do if a^2+b^2+2*c^2 > n then break; fi; for d from c+1 do if a^2+b^2+c^2+d^2 > n then break; elif a^2+b^2+c^2+d^2 = n then res := res+1 ; fi ; end do; end do; end do: end do: res ; end proc:
%p A175958 A := proc(n) A025443(n^2) ; end proc: seq(A(n),n=0..60) ;
%p A175958 # second Maple program:
%p A175958 b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
%p A175958       `if`(t*i^2<n, 0, `if`(i=1, 0, b(n,i-1,t))+
%p A175958       `if`(i^2>n, 0, b(n-i^2,i-1,t-1))))
%p A175958     end:
%p A175958 a:= n-> b(n^2, n, 4):
%p A175958 seq(a(n), n=0..80);  # _Alois P. Heinz_, Feb 07 2013
%t A175958 b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[t*i^2 < n, 0, If[i == 1, 0, b[n, i-1, t]] + If[i^2 > n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n^2, n, 4]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Jun 24 2015, after _Alois P. Heinz_ *)
%Y A175958 Cf. A004195, A004441, A004433, A181524.
%K A175958 nonn,look
%O A175958 0,14
%A A175958 _R. J. Mathar_, Oct 30 2010
%E A175958 More terms from _Alois P. Heinz_, Feb 07 2013