%I #35 Sep 20 2020 08:19:58
%S 1,4,4,8,12,12,12,16,16,20,32,24,20,36,28,32,48,32,36,48,40,44,48,48,
%T 56,64,52,48,80,60,60,96,48,68,96,72,72,80,96,80,108,84,64,112,88,96,
%U 128,80,96,144,100,104,128,108,108,144,112,96,144,128,132,160
%N a(n) is 1/6 of the number of primitive integral quadruples with sum = 2*m and sum of squares = 2*m^2, where m = 2*n-1.
%C Set b(m) = a(n) for m = 2*n-1, and b(m) = 0 for m even.
%C Conjecture: b(m) is multiplicative: for k >= 1, b(2^k) = 0; b(p^k) = p^(k-1)*b(p) for p an odd prime; b(p) = p+1 for p == 3 (mod 4); b(p) = p-1 for p == 1 (mod 4). It would be nice to have a proof of this.
%H Andrew Howroyd, <a href="/A278083/b278083.txt">Table of n, a(n) for n = 1..500</a>
%H Petros Hadjicostas, <a href="/A278081/a278081.r.txt">Slight modification of Mallows' R program</a>. [To get the total counts for n = 1 to 120, with the zeros, i.e., the sequence (b(n): n >= 1) shown in the comments above, type gc(1:120, 2, 2), where r = 2 and s = 2. To get the 1/6 of these counts with no zeros, type gc(seq(1,59,2), 2, 2)[,3]/6.]
%H Colin Mallows, <a href="/A278081/a278081_2.txt">R programs for A278081-A278086</a>.
%e 6*a(2) = 24 = 6*b(3) because of (-1,2,2,3) and (0,1,1,4) (12 permutations each). For example, (-1) + 2 + 2 + 3 = 6 = 2*3 and (-1)^2 + 2^2 + 2^2 + 3^2 = 18 = 2*3^2 (with n = 2 and m = 3 = 2*n - 1).
%t sqrtint = Floor[Sqrt[#]]&;
%t q[r_, s_, g_] := Module[{d = 2 s - r^2, h}, If[d <= 0, d == 0 && Mod[r, 2] == 0 && GCD[g, r/2] == 1, h = Sqrt[d]; If[IntegerQ[h] && Mod[r + h, 2]==0 && GCD[g, GCD[(r+h)/2, (r-h)/2]]==1, 2, 0]]] /. {True -> 1, False -> 0};
%t a[n_] := Module[{m = 2n - 1, s}, s = 2m^2; Sum[q[2m - i - j, s - i^2 - j^2, GCD[i, j]] , {i, -sqrtint[s], sqrtint[s]}, {j, -sqrtint[s - i^2], sqrtint[s - i^2]}]/6];
%t Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 100}] (* _Jean-François Alcover_, Sep 20 2020, after _Andrew Howroyd_ *)
%o (PARI)
%o q(r, s, g)={my(d=2*s - r^2); if(d<=0, d==0 && r%2==0 && gcd(g, r/2)==1, my(h); if(issquare(d, &h) && (r+h)%2==0 && gcd(g, gcd((r+h)/2, (r-h)/2))==1, 2, 0))}
%o a(n)={my(m=2*n-1, s=2*m^2); sum(i=-sqrtint(s), sqrtint(s), sum(j=-sqrtint(s-i^2), sqrtint(s-i^2), q(2*m-i-j, s-i^2-j^2, gcd(i,j)) ))/6} \\ _Andrew Howroyd_, Aug 02 2018
%Y Cf. A046897, A278081, A278082, A278084, A278085, A278086.
%K nonn
%O 1,2
%A _Colin Mallows_, Nov 14 2016
%E Terms a(51) and beyond from _Andrew Howroyd_, Aug 02 2018
%E Name and example section edited by _Petros Hadjicostas_, Apr 21 2020