OFFSET
1,1
COMMENTS
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).
FORMULA
T(n, 2*j-1) = x(n)_j, and T(n, 2*j) = y(n)_j, for j = 1, 2, ..., A336886(n).
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Wolfdieter Lang, Aug 10 2020
STATUS
approved