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A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.

%I A382407 #6 Apr 10 2025 21:13:21
%S A382407 0,0,0,0,0,1,1,0,0,0,1,1,0,3,0,1,1,1,2,2,2,1,1,1,1,3,0,5,1,1,2,3,3,2,
%T A382407 1,1,3,6,1,4,2,7,4,4,0,3,5,3,4,2,1,7,2,1,5,9,3,5,3,4,1,9,2,6,3,5,6,5,
%U A382407 4,7,5,1,5,6,3,13,7,8,4,6,0,4,4,11,5,13,2
%N A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.
%C A382407 a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.
%C A382407 Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.
%H A382407 Felix Huber, <a href="/A382407/b382407.txt">Table of n, a(n) for n = 1..10000</a>
%H A382407 Felix Huber, <a href="/A382407/a382407.txt">Maple program to calculate the partitions</a>
%e A382407 The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.
%p A382407 A382407:=proc(n)
%p A382407     local a,x,y,z;
%p A382407     a:=0;
%p A382407     for x to n/3 do
%p A382407         for y from x to (n-x)/2 do
%p A382407             z:=n-x-y;
%p A382407             if issqr(x*y+x*z+y*z) then
%p A382407                 a:=a+1
%p A382407             fi
%p A382407         od
%p A382407     od;
%p A382407     return a
%p A382407 end proc;
%p A382407 seq(A382407(n),n=1..87);
%Y A382407 Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.
%K A382407 nonn
%O A382407 1,14
%A A382407 _Felix Huber_, Apr 04 2025