A278084 a(n) is 1/24 of the number of primitive integral quadruples with sum = 2*m and sum of squares = 6*m^2, where m = 2*n-1.

1, 2, 5, 6, 6, 12, 14, 10, 18, 20, 12, 22, 25, 18, 28, 32, 24, 30, 38, 28, 40, 42, 30, 46, 42, 36, 54, 60, 40, 60, 60, 36, 70, 66, 44, 72, 74, 50, 72, 80, 54, 82, 90, 56, 88, 84, 64, 100, 98, 72, 100, 102, 60, 106, 108, 76, 114, 110, 84, 108, 132, 80, 125, 126

Offset: 1

Author

Colin Mallows, Nov 14 2016

Keywords

• nonn

Comments

Set b(m) = a(n) for m = 2*n-1, and b(m) = 0 for m even.

Conjecture: b(m) is multiplicative: for k >= 1, b(2^k) = 0, and for p an odd prime, b(p^k) = p^(k-1)*b(p), with b(p) = p + 1 for p == (11, 13, 17, 19) (mod 20), b(p) = p - 1 for p == (1, 3, 7, 9) (mod 20), b(5) = 5. It would be nice to have a proof of this.

Examples

			24*a(2) = 48 = 24*b(3) because of (-4,2,3,5) and (-2,0,1,7) (24 permutations each). For example, (-2) + 0 + 1 + 7 = 6 = 2*3 and (-2)^2 + 0^2 + 1^2 + 7^2 = 54 = 6*3^2 (with n = 2 and m = 3 = 2*2 - 1).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Petros Hadjicostas, Slight modification of Mallows' R program. [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, 6), where r = 2 and s = 6. To get the 1/24 of these counts with no zeros, type gc(seq(1,59,2), 2, 6)[,3]/24.]
• Colin Mallows, R programs for A278081-A278086.

Crossrefs

Cf. A046897, A278081, A278082, A278083, A278085, A278086.

Programs

Mathematica

sqrtint = Floor[Sqrt[#]]&;
q[r_, s_, g_] := Module[{d = 2s - 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};
a[n_] := Module[{m = 2n - 1, s}, s = 6m^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]}]/24];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Sep 20 2020, after Andrew Howroyd *)

PARI

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))}
a(n)={my(m=2*n-1, s=6*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)) ))/24} \\ Andrew Howroyd, Aug 02 2018

Extensions

Terms a(51) and beyond from Andrew Howroyd, Aug 02 2018

Name and example section edited by Petros Hadjicostas, Apr 21 2020