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A372217 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.

%I A372217 #28 Jun 17 2024 15:24:58
%S A372217 0,1,3,8,14,36,48,100,146,232,294,502,595,938,1143,1433,1741,2512,
%T A372217 2826,3911,4458,5319,6067,7976,8728,10750,12076,14194,15671,19510,
%U A372217 20669,25349,28115,31716,34697,39467,41894,49766,54046,59948,63951,74818,78216,90773,97220
%N A372217 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 A372217 Felix Huber, <a href="/A372217/a372217.pdf">Illustration of the terms a(1) to a(6)</a>.
%e A372217 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 A372217 S372217:=proc(n);
%p A372217   local s,x,u,v;
%p A372217   s:=0;
%p A372217   if n=1 then return 1 fi;
%p A372217   for x to n do
%p A372217     if gcd(x,n)=1 then
%p A372217       for u from x to n do
%p A372217         for v from 0 to n do
%p A372217           if gcd(u,v)=1 and gcd(u-x,n-v)=1 then
%p A372217             if u<n then s:=s+1;
%p A372217             elif v>=x then s:=s+1;
%p A372217             fi;
%p A372217           fi;
%p A372217         od;
%p A372217       od;
%p A372217     fi;
%p A372217   od;
%p A372217   return s;
%p A372217 end proc;
%p A372217 A372217:=proc(n)
%p A372217   local i,a;
%p A372217   a:=0;
%p A372217   for i from 0 to n do
%p A372217     a:=a+S372217(i);
%p A372217   od;
%p A372217   return a;
%p A372217 end proc;
%p A372217 seq(A372217(n),n=0..44);
%Y A372217 Cf. A115004, A141224, A141255, A320540, A320541, A320544, A372218.
%K A372217 nonn
%O A372217 0,3
%A A372217 _Felix Huber_, Apr 28 2024