A320541 Triangle read by rows: T(n,k) (1<=k<=n) = Sum_{i=1..n, j=1..k, gcd(i,j)=1} (n+1-i)*(k+1-j).

1, 3, 8, 6, 16, 31, 10, 26, 50, 80, 15, 39, 75, 120, 179, 21, 54, 103, 164, 244, 332, 28, 72, 137, 218, 324, 441, 585, 36, 92, 175, 278, 413, 562, 745, 948, 45, 115, 218, 346, 514, 699, 926, 1178, 1463, 55, 140, 265, 420, 623, 846, 1120, 1424, 1768, 2136

Offset: 1

Author

Hugo Pfoertner, Oct 15 2018

Keywords

• nonn
• tabl

Comments

T(n,k) = (1/4) * number of ways to select 3 distinct points forming a triangle of unsigned area = 1/2 from a rectangle of grid points with side lengths n and k.

Permutations of the 3 points are not counted separately.

Examples

			The triangle begins:
   1
   3   8
   6  16   31
  10  26   50   80
  15  39   75  120  179
  21  54  103  164  244  332
  28  72  137  218  324  441 585
...
a(1) = 1 because 4 triangles of area 1/2 in a [0 1]X[0 1] square can be formed by cutting the unit square into 2 triangles along the diagonals.
		

Links

• Seiichi Manyama, Rows n = 1..140, flattened

Crossrefs

Cf. A000217, A115004 (main diagonal), A320539, A320543, A333292.

This triangle is equivalent to the table in A114999.

Programs

Maple

T := proc(m,n) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i,j)=1 then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
for m from 1 to 12 do lprint([seq(T(m,n),n=1..m)]); od: # N. J. A. Sloane, Feb 04 2020

Extensions

Replaced definition (now a comment) by explicit formula. - N. J. A. Sloane, Feb 04 2020