%I #28 Jun 17 2024 15:24:58
%S 0,1,3,8,14,36,48,100,146,232,294,502,595,938,1143,1433,1741,2512,
%T 2826,3911,4458,5319,6067,7976,8728,10750,12076,14194,15671,19510,
%U 20669,25349,28115,31716,34697,39467,41894,49766,54046,59948,63951,74818,78216,90773,97220
%N a(n) is the number of distinct triangles whose sides do not pass through a grid point and whose vertices are three points of an n X n grid.
%H Felix Huber, <a href="/A372217/a372217.pdf">Illustration of the terms a(1) to a(6)</a>.
%e See the linked illustration for the terms a(1) = 1, a(2) = 3, a(3) = 8, a(4) = 14, a(5) = 36 and a(6) = 48.
%p S372217:=proc(n);
%p local s,x,u,v;
%p s:=0;
%p if n=1 then return 1 fi;
%p for x to n do
%p if gcd(x,n)=1 then
%p for u from x to n do
%p for v from 0 to n do
%p if gcd(u,v)=1 and gcd(u-x,n-v)=1 then
%p if u<n then s:=s+1;
%p elif v>=x then s:=s+1;
%p fi;
%p fi;
%p od;
%p od;
%p fi;
%p od;
%p return s;
%p end proc;
%p A372217:=proc(n)
%p local i,a;
%p a:=0;
%p for i from 0 to n do
%p a:=a+S372217(i);
%p od;
%p return a;
%p end proc;
%p seq(A372217(n),n=0..44);
%Y Cf. A115004, A141224, A141255, A320540, A320541, A320544, A372218.
%K nonn
%O 0,3
%A _Felix Huber_, Apr 28 2024