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A332595
Number of triangular regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 4.
3
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from N. J. A. Sloane, Mar 09 2020
STATUS
approved