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A336887
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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.
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4
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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, 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, 6, 3, 1, 7, 5, 10, 11, 8
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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The irregular triangle T(n, j) begins:
n, z(n) \ j 1 2 3 4 5 6 7 8 9 10 ...
--------------------------------------------------------------------------
1, 5: 2
2, 8: 1
3, 9: 3
4, 10: 1 3 4
5, 13: 2 1
6, 15: 6
7, 16: 2 2 4 6
8, 17: 1 3 5 4
9, 18: 3 6
10, 20: 2 3 1 1 4 2 6 7 6 8
11, 24: 3
12, 25: 1 3 5 4 8 2 10
13, 26: 1 4 2 3 7 9 5
14, 27: 9
15, 29: 3 2 4 1 8 9 7 10
16, 30: 3 9 12
17, 32: 4 2 6 4 2 10 8 12 6
...
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.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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