A146557 Number of collinear triples of distinct points in Zn x Zn with no two on the same "horizontal" or "vertical" line.

0, 0, 6, 32, 200, 384, 1470, 2688, 5400, 9600, 18150, 27168, 44616, 65856, 90150, 140800, 184960, 274320, 331398, 474400, 569184, 774400, 896126, 1366656, 1390000, 1881984, 2204982, 2899232, 2967048, 4545600, 4180350, 5904384

Offset: 1

Author

Max Alekseyev, Oct 31 2008

Keywords

• nonn

Comments

Number of 3x3 matrices [1, x, u; 1, y, v; 1, z, w] over Z_n with zero determinant, where elements of the triples x,y,z and u,v,w are distinct.

Conjecture: if 3 does not divide n then n^2 divides a(n). - Peter Bala, Jul 24 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Programs

Mathematica

f[n_] := n*Sum[ Sum[ (n - i - j)*( n*GCD[i, j, n - i - j] - GCD[i, n] - GCD[j, n] - GCD[i + j, n] + 2 ) , {j, 1, n - i}] , {i, 1, n}]; Table[f[n], {n,1,25}] (* G. C. Greubel, Oct 18 2016 *)

PARI

{ a(n) = n * sum(i=1,n, sum(j=1,n-i, (n-i-j) * (n*gcd([i,j,n-i-j]) - gcd(i,n) - gcd(j,n) - gcd(i+j,n) + 2) )) }

Formula

a(n) = n * Sum_{i,j,k} ( n * gcd(i,j,k) - gcd(i,n) - gcd(j,n) - gcd(k,n) + 2 ) * k, where the sum is taken over all triples of positive integers i,j,k with i+j+k=n.