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 A146557 Number of collinear triples of distinct points in Zn x Zn with no two on the same "horizontal" or "vertical" line. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 A221540 A121120 Adjacent sequences:  A146554 A146555 A146556 * A146558 A146559 A146560 KEYWORD nonn AUTHOR Max Alekseyev, Oct 31 2008 STATUS approved

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