A135538 Number of triples in all permutations of order n that are collinear modulo n.

0, 0, 0, 6, 32, 400, 2304, 35280, 322560, 3888000, 48384000, 731808000, 9858723840, 161902540800, 2628760780800, 43181994240000, 876764528640000, 16124496740352000, 358721232629760000, 6933770723303424000, 168738115888742400000, 3644128675321085952000, 94201965756599500800000

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

L. Li, Collinear triples in permutations, arXiv:0802.0572 [math.CO], 2008.

Formula

For n>=3, a(n) = (n-3)! * A146557(n).

Example

For example, in a permutation p=[5,2,4,3,1], a triple of points { (2,p(2)=2), (4,p(4)=3), (5,p(5)=1) } is collinear, since they are located on the line: x + 3*y == 3 (mod 5).

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}]; Join[{0, 0, 0}, Table[(n - 3)!*f[n], {n, 3, 25}]] (* G. C. Greubel, Oct 18 2016 *)

Prog

(PARI) a(n) = if(n<3, 0, (n-3)! * n * sum(i=1, n, sum(j=1, n-i-1, (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: A185386 A368287 A276351 * A296314 A222605 A290790

Adjacent sequences: A135535 A135536 A135537 * A135539 A135540 A135541

Keyword

nonn

Author

R. J. Mathar, Feb 22 2008, corrected Oct 24 2008

Extensions

Edited and extended by Max Alekseyev, Oct 31 2008

Status

approved