A337216 Irregular triangle read by rows: a(n, j) gives the positive integer area A(n)_j corresponding to the nonrectangular triangles with sides (sqrt(x(n)_j), sqrt(y(n)_j), z(n)), with integers 1 <= x(n)_j <= y(n)_j <= z(n), that lead to primitive quartets (x(n)_j, y(n)_j, z(n), a(n, j)). Hence x(n)_j = A336888(n, 2*j-1), y(n)_j = A336888(n, 2*j), for j = 1, 2, ..., A336889(n), and z(n) = A337215(n), for n >= 1.

2, 1, 3, 1, 3, 2, 1, 2, 4, 6, 1, 3, 5, 4, 3, 3, 1, 1, 4, 2, 6, 7, 1, 3, 5, 4, 8, 2, 1, 3, 7, 9, 5, 3, 2, 4, 1, 8, 9, 7, 10, 2, 6, 2, 10, 6, 3, 1, 7, 5, 11, 3, 6, 3, 9, 12, 3, 6, 9, 15, 2, 1, 5, 4, 8, 7, 11, 10, 14, 6, 3, 1, 7, 9, 1, 11, 13, 3, 9, 1, 8, 3, 6, 4, 5, 2, 12, 15, 13, 7, 11, 17

Offset

1,1

Links

Table of n, a(n) for n=1..92.

Formula

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, ..., A336889(n), with x(n)_j = A336888(n, 2*j-1), y(n)_j = A336888(n, 2*j) and z(n) = A337215(n), for n >= 1.

Example

The irregular triangle a(n,j) begins (z(n) = A337215(n)):
n, z(n) \ j  1 2  3  4  5  6  7  8  9 10 11 12 13 ...
-----------------------------------------------------
1,   5:      2
2,   8:      1
3,   9:      3
4,  10:      1 3
5,  13:      2 1
6,  16:      2 4  6
7,  17:      1 3  5  4
8,  18:      3
9,  20:      3 1  1  4  2  6  7
10, 25:      1 3  5  4  8  2
11, 26:      1 3  7  9  5
12, 29:      3 2  4  1  8  9  7 10
13, 32:      2 6  2 10  6
14, 34:      3 1  7  5 11
15, 36:      3 6  3  9 12  3  6  9 15
16: 37:      2 1  5  4  8  7 11 10 14  6
17, 40:      3 1  7  9  1 11 13  3  9
18, 41:      1 8  3  6  4  5  2 12 15 13  7 11 17
19, 45:      6 3  9  3 12  6  6 12  3  9
20, 49:      7 7 14  7 14
21, 50:      9 1  5 13  3 15  7
...
-----------------------------------------------------
a(6, 3) = 6 for the triangle from row n = 6 of A336888: (x(6)_5 , y(6)_6, z(6))  = (13, 13, 16), with area (in some square length units) (1/4)*sqrt(2*(16*13 + 16*13 + 13*13) - (2*13^2 +16^2)) = 6.

Crossrefs

Cf. A334818, A336885, A336885, A336887, A336889 (row lengths), A337215 (z(n)), A337216 (areas).

Sequence in context: A083868 A128199 A345063 * A249617 A304091 A278801

Adjacent sequences: A337213 A337214 A337215 * A337217 A337218 A337219

Keyword

nonn,tabf

Author

Wolfdieter Lang, Aug 19 2020

Status

approved