A241222 Number of collinear point triples on a centered hexagonal grid of size n.

0, 3, 69, 390, 1314, 3441, 7503, 14388, 25692, 42471, 66417, 100194, 145206, 204429, 280971, 377400, 496608, 642891, 821925, 1034742, 1288602, 1587009, 1933695, 2339100, 2802804, 3334983, 3942585, 4627002, 5404542, 6278661, 7252539, 8332968, 9537456

Offset: 1

Author

Martin Renner, Apr 17 2014

Keywords

• nonn

Comments

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

Examples

			For n = 2 the points are on the three diagonals through the center of the hexagon as following:
    . .     . *     * .
   * * *   . * .   . * .
    . .     * .     . *
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Hex Number.

Crossrefs

Cf. A000938, A241220.

Programs

PARI

c(n,s,fmin,fmax)={sum(k=1+s, n, max(0, fmax(k-s)-max(fmin(k)-1,if(k-2*s>0,fmax(k-2*s)))))}
b(n, u, v)={c(2*n-1, u, i->max(0,i-n)+1+i\u*v, i->min(i,n)+n-1+i\u*v)}
gm(n)={my(v=vector(n)); for(g=2, n, v[g]=binomial(g+1, 3) - sum(k=2, g-1, v[k]*min(k, g-k+1))); v}
a(n)={my(gmv=gm(n-1)); 3*(binomial(2*n-1,3) + 2*sum(k=0,n-2,binomial(n+k,3)) + sum(u=1, 2*n-3, sum(v=1, 2*n-2-u, my(m=gmv[gcd(u,v)]);if(m>0, m*b(n,u,v), 0))))} \\ Andrew Howroyd, Sep 18 2017

Extensions

a(7) from Martin Renner, May 31 2014

a(8)-a(22) from Giovanni Resta, May 31 2014

Terms a(23) and beyond from Andrew Howroyd, Sep 18 2017