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

1,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Hex Number.

Example

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

Prog

(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

Crossrefs

Cf. A000938, A241220.

Sequence in context: A046432 A232326 A270869 * A166835 A166806 A219070

Adjacent sequences: A241219 A241220 A241221 * A241223 A241224 A241225

Keyword

nonn

Author

Martin Renner, Apr 17 2014

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

Status

approved