cp's OEIS Frontend

A332595 Number of triangular regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 4.

%I A332595 #25 Mar 09 2020 22:03:42
%S A332595 1,12,32,72,128,232,368,576,832,1232,1712,2328,3040,4040,5184,6616,
%T A332595 8224,10248,12496,15144,18048,21688,25664,30184,35072,41000,47392,
%U A332595 54608,62336,71088,80416,90864,101952,114832,128480,143352,159040,176984,195888,216424,237984,261624,286384,313184,341184,372496,405184
%N A332595 Number of triangular regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 4.
%C A332595 See A331776 for many other illustrations.
%C A332595 Theorem. Let z_2(n) = Sum_{i, j = 1..n, gcd(i,j)=2} (n+1-i)*(n+1-j) (this is A331761). Then, for n >= 2, a(n) = 2*(z_2(n) + (n+3)*(n-1)). - _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 06 2020
%H A332595 Scott R. Shannon, <a href="/A331776/a331776.png">Colored illustration for a(3) = 32</a> (there are 4*32 triangles).
%p A332595 V := proc(m,n,q) local a,i,j; a:=0;
%p A332595 for i from 1 to m do for j from 1 to n do
%p A332595 if gcd(i,j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
%p A332595 f := n -> if n=1 then 4 else 8*n^2 + 16*n - 24 + 8*V(n,n,2); fi;
%p A332595 [seq(f(n)/4, n=1..60)]; # _N. J. A. Sloane_, Mar 09 2020
%Y A332595 Cf. A331761, A331776, A332594, A332596.
%K A332595 nonn
%O A332595 1,2
%A A332595 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 02 2020
%E A332595 More terms from _N. J. A. Sloane_, Mar 09 2020