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 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. 11
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 LINKS Table of n, a(n) for n=1..65. 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 EXAMPLE 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] ... 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 *) CROSSREFS Cf. A114043, A115004 (main diagonal), A115005, A115006, A115007, A320541. Sequence in context: A360375 A098832 A107985 * A160733 A021752 A049626 Adjacent sequences: A114996 A114997 A114998 * A115000 A115001 A115002 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Feb 23 2006 STATUS approved

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Last modified December 8 14:30 EST 2023. Contains 367679 sequences. (Running on oeis4.)