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

1,2

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

Table of n, a(n) for n=1..47.

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

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

Crossrefs

Cf. A331761, A331776, A332594, A332596.

Sequence in context: A134582 A177721 A081268 * A188183 A183682 A243027

Adjacent sequences: A332592 A332593 A332594 * A332596 A332597 A332598

Keyword

nonn

Author

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

Extensions

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

Status

approved