A114999 Array read by antidiagonals: T(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)*(n+1-j), m>=1, n>=1.

1, 3, 3, 6, 8, 6, 10, 16, 16, 10, 15, 26, 31, 26, 15, 21, 39, 50, 50, 39, 21, 28, 54, 75, 80, 75, 54, 28, 36, 72, 103, 120, 120, 103, 72, 36, 45, 92, 137, 164, 179, 164, 137, 92, 45, 55, 115, 175, 218, 244, 244, 218, 175, 115, 55, 66, 140, 218, 278, 324, 332, 324, 278, 218, 140

Offset: 1

Author

N. J. A. Sloane, Feb 23 2006

Keywords

• nonn
• tabl

Comments

The corresponding triangle is A320541, counting (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 j and k, written as triangle T(j,k), j<=k. - Hugo Pfoertner, Oct 22 2018

Examples

			The top left corner of the array is:
[1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78]
[3, 8, 16, 26, 39, 54, 72, 92, 115, 140, 168, 198]
[6, 16, 31, 50, 75, 103, 137, 175, 218, 265, 318, 374]
[10, 26, 50, 80, 120, 164, 218, 278, 346, 420, 504, 592]
[15, 39, 75, 120, 179, 244, 324, 413, 514, 623, 747, 877]
[21, 54, 103, 164, 244, 332, 441, 562, 699, 846, 1014, 1190]
[28, 72, 137, 218, 324, 441, 585, 745, 926, 1120, 1342, 1575]
[36, 92, 175, 278, 413, 562, 745, 948, 1178, 1424, 1706, 2002]
[45, 115, 218, 346, 514, 699, 926, 1178, 1463, 1768, 2118, 2485]
[55, 140, 265, 420, 623, 846, 1120, 1424, 1768, 2136, 2559, 3002]
[66, 168, 318, 504, 747, 1014, 1342, 1706, 2118, 2559, 3065, 3595]
[78, 198, 374, 592, 877, 1190, 1575, 2002, 2485, 3002, 3595, 4216]
...
		

Links

• Max A. Alekseyev. On the number of two-dimensional threshold functions. SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

Crossrefs

Cf. A114043, A115004 (main diagonal), A115005, A115006, A115007, A320541.

Programs

Maple

T:=proc(m,n) local t1,i,j; t1:=0; for i from 1 to m do for j from 1 to n do if gcd(i,j)=1 then t1:=t1+(m+1-i)*(n+1-j); fi; od; od; t1; end;

Mathematica

T[m_, n_] := Module[{t1, i, j}, t1 = 0; For[i = 1, i <= m, i++, For[j = 1, j <= n, j++, If[GCD[i, j] == 1 , t1 = t1 + (m+1-i)*(n+1-j)]]]; t1]; Table[T[m-n+1, n], {m, 1, 11}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 07 2014, translated from Maple *)