A372218 - OEIS

%I #21 Jun 17 2024 15:25:29

%S 0,4,36,184,592,1828,4164,9360,18592,34948,59636,102096,161496,255700,

%T 385292,562336,796344,1131996,1552780,2133368,2855632,3765492,4876444,

%U 6328104,8049744,10203820,12766508,15870744,19496392,23984444,29090340,35318968,42535496,50936036

%N a(n) is the number of ways to select three distinct points of an n X n grid forming a triangle whose sides do not pass through a grid point.

%C a(n) is 1/6 of the number of ways to select three points (x,y), (u,v), (p,q) with gcd(x-u,y-v) = gcd(u-p,v-q) = gcd(p-x,q-y) = 1 and 0 <= x, y, u, v, p, q <= n in an n X n grid.

%H Felix Huber, <a href="/A372218/a372218.pdf">Illustration of a(2)</a>

%e See the linked illustration: a(2) = 36 because there are 36 ways to select three distinct points in a square grid with side length n that satisfy the condition.

%p A372218:=proc(n)

%p   local x,y,u,v,p,q,a;

%p   a:=0;

%p   for x from 0 to n do

%p     for y from 0 to n do

%p       for u from 0 to n do

%p         for v from 0 to n do

%p           if gcd(x-u,y-v)=1 then

%p             for p from 0 to n do

%p               for q from 0 to n do

%p                 if gcd(x-p,y-q)=1 and gcd(p-u,q-v)=1 then a:=a+1 fi;

%p               od;

%p             od;

%p           fi;

%p         od;

%p       od;

%p     od;

%p   od;

%p   a:=a/6;

%p   return a;

%p end proc;

%p seq(A372218(n),n=0..33);

%Y Cf. A115004, A141224, A141255, A320540, A320541, A320544, A372217.

%K nonn

%O 0,2

%A _Felix Huber_, Apr 28 2024