A114999 - OEIS

%I #20 May 28 2020 16:25:17

%S 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,

%T 75,54,28,36,72,103,120,120,103,72,36,45,92,137,164,179,164,137,92,45,

%U 55,115,175,218,244,244,218,175,115,55,66,140,218,278,324,332,324,278,218,140

%N 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.

%C 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

%H Max A. Alekseyev. <a href="http://arXiv.org/abs/math.CO/0602511">On the number of two-dimensional threshold functions</a>. SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

%e The top left corner of the array is:

%e [1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78]

%e [3, 8, 16, 26, 39, 54, 72, 92, 115, 140, 168, 198]

%e [6, 16, 31, 50, 75, 103, 137, 175, 218, 265, 318, 374]

%e [10, 26, 50, 80, 120, 164, 218, 278, 346, 420, 504, 592]

%e [15, 39, 75, 120, 179, 244, 324, 413, 514, 623, 747, 877]

%e [21, 54, 103, 164, 244, 332, 441, 562, 699, 846, 1014, 1190]

%e [28, 72, 137, 218, 324, 441, 585, 745, 926, 1120, 1342, 1575]

%e [36, 92, 175, 278, 413, 562, 745, 948, 1178, 1424, 1706, 2002]

%e [45, 115, 218, 346, 514, 699, 926, 1178, 1463, 1768, 2118, 2485]

%e [55, 140, 265, 420, 623, 846, 1120, 1424, 1768, 2136, 2559, 3002]

%e [66, 168, 318, 504, 747, 1014, 1342, 1706, 2118, 2559, 3065, 3595]

%e [78, 198, 374, 592, 877, 1190, 1575, 2002, 2485, 3002, 3595, 4216]

%e ...

%p 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;

%t 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 *)

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

%K nonn,tabl

%O 1,2

%A _N. J. A. Sloane_, Feb 23 2006