A278147 Triangle read by rows of Cantor pairing function value determining primitive Pythagorean triangles or 0 if there is no such triangle.

8, 0, 18, 19, 0, 32, 0, 33, 0, 50, 34, 0, 0, 0, 72, 0, 52, 0, 73, 0, 98, 53, 0, 74, 0, 99, 0, 128, 0, 75, 0, 100, 0, 0, 0, 162, 76, 0, 101, 0, 0, 0, 163, 0, 200, 0, 102, 0, 131, 0, 164, 0, 201, 0, 242, 103, 0, 0, 0, 165, 0, 202, 0, 0, 0, 288, 0, 133, 0, 166, 0, 203, 0, 244, 0, 289, 0, 338, 134, 0, 167, 0, 204

Offset: 2

Author

Wolfdieter Lang, Nov 21 2016

Keywords

• nonn
• tabl
• easy

Comments

This entry is inspired by the increasingly ordered nonvanishing entries given in A277557.

A primitive Pythagorean triangle is characterized by the pair [n,m], 1 <= m < n, GCD(n,m) = 1 and n+m is odd. The present triangle gives the values T(n, m) = Cantor(m,n) where Cantor(x,y) = (x+y)*(x+y+1)/2 + y. See A277557, also for links.

Because the Cantor pairing function N x N -> N is bijective (N = positive integers), all nonzero entries of this triangle appear only once, but here not all positive integers appear.

Note that in this triangle in each row the nonvanishing entries increase, but in the first rows up to some n not all T(n, m) values smaller than T(n,n-1) are covered.

For the area values of primitive Pythagorean triangles see the table A249869 also for comments on these triangles and references.

Examples

			The triangle begins:
n\m  1   2   3   4   5   6   7   8   9  10...
2:  8
3:  0   18
4:  19   0  32
5:   0  33   0  50
6:  34   0   0   0  7272
7:   0  52   0  73   0  98
8:  53   0  74   0  99   0 128
9:   0  75   0 100   0   0   0 162
10: 76   0 101   0   0   0 163   0 200
11:  0 102   0 131   0 164   0 201   0 242
...
n = 12: 103 0 0 0 165 0 202 0 0 0 288,
n = 13: 0 133 0 166 0 203 0 244 0 289 0 338,
n = 14: 134 0 167 0 204 0 0 0 290 0 339 0 392,
n = 15: 0 168 0 205 0 0 0 291 0 0 0 0 0 450.
...
T(3,1) = 0 because 3+1 =4 is even.
T(4,2) = 0 because GCD(4,2) = 2 > 1.
T(3,2) = (2+3)*(2+3)/2 + 3 = 5*3 + 3 = 18.
...
In order to reach all values T(n,m) <= 50 one has to take rows n = 2..6.
...
		

Crossrefs

Cf. A277557, A249869.

Formula

T(n, m) = (m+n)*(m+n+1)/2 + n, n >= 2, m = 1, 2, ..., n-1, and 0 if GCD(n,m) > 1 or n+m is even.