A189346 Number of sets of four points on an n X n grid (or geoboard), exactly three of which are collinear.

0, 0, 48, 532, 3088, 11340, 33824, 83288, 183344, 364304, 681872, 1194100, 1992976, 3182332, 4941360, 7420640, 10874720, 15539952, 21812720, 30011924, 40650368, 54187196, 71463440, 92990296, 119675712, 152314920, 192393872, 240690060

Offset: 1

Author

Martin Renner, Apr 20 2011

Keywords

• nonn

Comments

The four points build a triangle on an n X n grid, with one of them located on a side of the triangle.

The number of sets of four points with the three collinear points in a horizontal or vertical line is 2*n^2*(n-1)*binomial(n,3) = 4*A090448(n). The number of sets of four points with the three collinear points in a diagonal line of slope 1 is 2*n*(n-1)*binomial(n,3) + 4*Sum_{k=3..n-1}(n^2-k)*binomial(k,3). The sum of these two values is a lower bound for this sequence. - Nathaniel Johnston, Apr 23 2011

Links

• Martin Renner, Table of n, a(n) for n = 1..76

Crossrefs

Cf. A000938, A175383, A178256, A189345.

Programs

Maple

A189346 := proc(n)local a,b,j,k,l,m,s,slopes,num,den,tot: tot := 0: slopes := {}: for b from 1 to ceil(n/2)-1 do for a from 0 to b do slopes := slopes union {a/b}: od: od: for s from 1 to nops(slopes) do num := numer(slopes[s]): den := denom(slopes[s]): if(num = 0)then tot := tot + 2*n^2*(n-1)*binomial(n,3): elif(num = den)then tot := tot + 2*(2*add(binomial(k,3)*(n^2-k), k=3..n) - binomial(n,3)*(n^2 - n)): else for j from 1 to n - 2*den do for k from 1 to n - 2*num do tot := tot + 4*(n^2 - 3): for l from 1 to n do for m from 1 to n do if((not l = j or not m = k) and (not l = j + den or not m = k + num) and (not l = j + 2*den or not m = k + 2*num) and (m - k)*den = num*(l - j))then tot := tot - 4: fi: od: od: od: od: fi: od: return tot: end:
seq(A189346(n),n=1..15); # Nathaniel Johnston, Apr 23 2011

Extensions

a(6)-a(28) from Nathaniel Johnston, Apr 23 2011