login
Number of collinear point triples on a centered hexagonal grid of size n.
2

%I #13 Sep 19 2017 03:42:46

%S 0,3,69,390,1314,3441,7503,14388,25692,42471,66417,100194,145206,

%T 204429,280971,377400,496608,642891,821925,1034742,1288602,1587009,

%U 1933695,2339100,2802804,3334983,3942585,4627002,5404542,6278661,7252539,8332968,9537456

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

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

%H Andrew Howroyd, <a href="/A241222/b241222.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexNumber.html">Hex Number</a>.

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

%e . . . * * .

%e * * * . * . . * .

%e . . * . . *

%o (PARI)

%o 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)))))}

%o 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)}

%o 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}

%o 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

%Y Cf. A000938, A241220.

%K nonn

%O 1,2

%A _Martin Renner_, Apr 17 2014

%E a(7) from _Martin Renner_, May 31 2014

%E a(8)-a(22) from _Giovanni Resta_, May 31 2014

%E Terms a(23) and beyond from _Andrew Howroyd_, Sep 18 2017