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

1, 12, 32, 72, 128, 232, 368, 576, 832, 1232, 1712, 2328, 3040, 4040, 5184, 6616, 8224, 10248, 12496, 15144, 18048, 21688, 25664, 30184, 35072, 41000, 47392, 54608, 62336, 71088, 80416, 90864, 101952, 114832, 128480, 143352, 159040, 176984, 195888, 216424, 237984, 261624, 286384, 313184, 341184, 372496, 405184

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 02 2020

Keywords

• nonn

Comments

See A331776 for many other illustrations.

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

Links

• Scott R. Shannon, Colored illustration for a(3) = 32 (there are 4*32 triangles).

Crossrefs

Cf. A331761, A331776, A332594, A332596.

Programs

Maple

V := proc(m,n,q) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i,j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
f := n -> if n=1 then 4 else 8*n^2 + 16*n - 24 + 8*V(n,n,2); fi;
[seq(f(n)/4, n=1..60)]; # N. J. A. Sloane, Mar 09 2020

Extensions

More terms from N. J. A. Sloane, Mar 09 2020