A336885 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.

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

Offset

1,1

Comments

This entry, A336886 and A336887 are inspired by A334818.

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).

Links

Table of n, a(n) for n=1..74.

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

Cf. A334818, A336886, A336887.

Sequence in context: A131131 A073241 A336888 * A094642 A069284 A272638

Adjacent sequences: A336882 A336883 A336884 * A336886 A336887 A336888

Keyword

nonn,tabf

Author

Wolfdieter Lang, Aug 10 2020

Status

approved