A336887 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.

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

Offset

1,1

Comments

The length of row n is A336886(n).

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.

Links

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

Formula

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).

Example

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.

Crossrefs

Cf. A334818, A336885, A336886.

Sequence in context: A213712 A143802 A177040 * A163313 A337178 A321893

Adjacent sequences: A336884 A336885 A336886 * A336888 A336889 A336890

Keyword

nonn,tabf

Author

Wolfdieter Lang, Aug 10 2020

Status

approved