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%I #8 Aug 11 2020 04:18:19
%S 2,1,3,1,3,4,2,1,6,2,2,4,6,1,3,5,4,3,6,2,3,1,1,4,2,6,7,6,8,3,1,3,5,4,
%T 8,2,10,1,4,2,3,7,9,5,9,3,2,4,1,8,9,7,10,3,9,12,4,2,6,4,2,10,8,12,6,2,
%U 6,3,1,7,5,10,11,8
%N Irregular triangle T(n, j) giving in row n the positive integer areas of all non-right angle triangles (X(n)_j, Y(n)_j, Z(n)), with X(n)_j = sqrt(x(n)_j), Y(n)_j = sqrt(y(n)_j), and Z(n) = sqrt(z(n)), and positive integers 1 <= x(n)_j <= y(n)_j <= z(n), for j = 1, 2,..., A336886. hence z(n) = A334818(n), for n >= 1.
%C The length of row n is A336886(n).
%C A(n)_j = (1/4)*sqrt(2*(z(n)*y(n)_j + z(n)*x(n)_j + y(n)_j*x(n)_j) - ((x(n)_j)^2 + (y(n)_j)^2 + z(n)^2)), for j = 1, 2, ..., A336886(n), with x(n)_j = A336885(n, 2*j-1), y(n)_j = A336885(n, 2*j), z(n) = A334818(n), for j = 1, 2, ..., A336886(n), for n >= 1.
%F For T(n, j), n >= 1, j = 1, 2, ..., A336886(n), see also the rows n of A336885 with the pairs (x(n)_j, y(n)_j).
%e The irregular triangle T(n, j) begins:
%e n, z(n) \ j 1 2 3 4 5 6 7 8 9 10 ...
%e --------------------------------------------------------------------------
%e 1, 5: 2
%e 2, 8: 1
%e 3, 9: 3
%e 4, 10: 1 3 4
%e 5, 13: 2 1
%e 6, 15: 6
%e 7, 16: 2 2 4 6
%e 8, 17: 1 3 5 4
%e 9, 18: 3 6
%e 10, 20: 2 3 1 1 4 2 6 7 6 8
%e 11, 24: 3
%e 12, 25: 1 3 5 4 8 2 10
%e 13, 26: 1 4 2 3 7 9 5
%e 14, 27: 9
%e 15, 29: 3 2 4 1 8 9 7 10
%e 16, 30: 3 9 12
%e 17, 32: 4 2 6 4 2 10 8 12 6
%e ...
%e T(7, 3) = 4 because the corresponding triangle has sides (X(7)_3, Y(7)_3, Z(7)_3) = (sqrt(x(7)_3), sqrt(x(7)_3), sqrt(z(7))), with x(7)_3 = A336885(7, 2*3-1) = 5, y(7)_3 = A336885(7, 2*3) = 13, z(7) = A334818(7) = 16, with area A(7)_3 = T(7, 3) = (1/4)*sqrt(2*(16*5 + 16*13 + 5*13) - (5^2 + 13^2 + 16^2)) = 4.
%Y Cf. A334818, A336885, A336886.
%K nonn,tabf
%O 1,1
%A _Wolfdieter Lang_, Aug 10 2020
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