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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).

%I A320541 #20 Aug 17 2021 10:13:33
%S A320541 1,3,8,6,16,31,10,26,50,80,15,39,75,120,179,21,54,103,164,244,332,28,
%T A320541 72,137,218,324,441,585,36,92,175,278,413,562,745,948,45,115,218,346,
%U A320541 514,699,926,1178,1463,55,140,265,420,623,846,1120,1424,1768,2136
%N 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).
%C A320541 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.
%C A320541 Permutations of the 3 points are not counted separately.
%H A320541 Seiichi Manyama, <a href="/A320541/b320541.txt">Rows n = 1..140, flattened</a>
%e A320541 The triangle begins:
%e A320541    1
%e A320541    3   8
%e A320541    6  16   31
%e A320541   10  26   50   80
%e A320541   15  39   75  120  179
%e A320541   21  54  103  164  244  332
%e A320541   28  72  137  218  324  441 585
%e A320541 ...
%e A320541 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.
%p A320541 T := proc(m,n) local a,i,j; a:=0;
%p A320541 for i from 1 to m do for j from 1 to n do
%p A320541 if gcd(i,j)=1 then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
%p A320541 for m from 1 to 12 do lprint([seq(T(m,n),n=1..m)]); od: # _N. J. A. Sloane_, Feb 04 2020
%Y A320541 Cf. A000217, A115004 (main diagonal), A320539, A320543, A333292.
%Y A320541 This triangle is equivalent to the table in A114999.
%K A320541 nonn,tabl
%O A320541 1,2
%A A320541 _Hugo Pfoertner_, Oct 15 2018
%E A320541 Replaced definition (now a comment) by explicit formula. - _N. J. A. Sloane_, Feb 04 2020