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

1,3

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.

Links

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

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.

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 *)

Prog

(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) )) }

Crossrefs

Sequence in context: A216441 A108188 A020058 * A020013 A283326 A221540

Adjacent sequences: A146554 A146555 A146556 * A146558 A146559 A146560

Keyword

nonn

Author

Max Alekseyev, Oct 31 2008

Status

approved