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A336885
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Irregular triangle read by rows: row n gives the sequence of pairs ((x(n)_j, y(n)_j))_{j=1..A336886(n)}, for all non-Pythagorean triples (X(n)_j = sqrt(x(n)_j), Y(n)_j = sqrt(y(n)_j), Z(n) = sqrt(z(n))), with integers 1 <= x(n)_j <= y(n)_j <= z(n) = A334818(n), which correspond to non-right triangles with positive integer area A(n)_j with leg pairs (X(n)_j, Y(n)_j) and base Z(n), for j = 1, 2, ..., A336886(n). The order of the pairs in row n is by nondecreasing y(n)_j.
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4
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4, 5, 1, 5, 5, 8, 2, 4, 4, 10, 8, 10, 4, 5, 1, 8, 12, 15, 5, 5, 2, 10, 5, 13, 13, 13, 4, 5, 5, 8, 8, 13, 4, 17, 4, 10, 10, 16, 4, 8, 5, 9, 2, 10, 1, 13, 5, 13, 1, 17, 9, 17, 13, 17, 8, 20, 16, 20, 3, 15, 5, 8, 4, 13, 8, 13, 4, 17, 16, 17, 1, 20, 20, 25
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OFFSET
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1,1
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COMMENTS
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The length of row n is 2*A336886(n).
The positive integer areas are 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). The corresponding irregular triangle of these areas is given in A336887. (Degenerate triangles are not considered, hence X(n)_j + Y(n)_j > Z(n).)
The triples (x(n)_j, y(n)_j, z(n)) may be non-primitive.
The positive or negative defect to Pythagorean triples is deltaPT(n)_j := (X(n)_j)^2 + (Y(n)_j)^2 - Z(n)^2 > -2*X(n)_j*Y(n)_j, for j = 1, 2, ..., A336886(n).
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LINKS
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FORMULA
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T(n, 2*j-1) = x(n)_j, and T(n, 2*j) = y(n)_j, for j = 1, 2, ..., A336886(n).
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EXAMPLE
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The irregular triangle T(n, k) with pairs (x(n)_j, y(n)_j), j= 1, 2, ..., A336886(n), begins:
n, z(n) \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 18 20 ...
--------------------------------------------------------------------------
1, 5: 4 5
2, 8: 1 5
3, 9: 5 8
4, 10: 2 4 4 10 8 10
5, 13: 4 5 1 8
6, 15: 12 15
7, 16: 5 5 2 10 5 13 13 13
8, 17: 4 5 5 8 8 13 4 17
9, 18: 4 10 10 16
10, 20: 4 8 5 9 2 10 1 13 5 13 1 17 9 17 13 17 8 20 16 20
11, 24: 3 15
12, 25: 5 8 4 13 8 13 4 17 16 17 1 20 20 25
13, 26: 4 10 8 10 2 16 2 20 10 20 18 20 4 26
14, 27: 15 24
15, 29: 8 9 4 13 5 16 1 20 13 20 17 20 8 25 16 29
16, 30: 6 12 12 30 24 30
17, 32: 10 10 5 13 9 17 4 20 1 25 17 25 10 26 26 26 5 29
...
T(4, 1) = 2 and T(4, 2) = 4 because for n = 4, z(4) = 10, k = 1: (X(10)_1 = sqrt(2), Y(10)_1 = sqrt(4)) and Z(n) = sqrt(10). This is not a Pythagorean triangle because 2 + 4 is not 10, and the area is integer: A(4, 1) = A336887(4, 1) = 1 because A(4, 1) = (1/4)*sqrt(2*(10*4 + 10*2 + 4*2) - (2^2 + 4^2 + 10^2)) = 1.
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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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