A321785 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt(A321769(n, k)^2 - A321768(n, k)^2).

1, 1, 3, 3, 1, 5, 5, 3, 7, 7, 3, 5, 5, 1, 7, 7, 5, 11, 11, 5, 9, 9, 3, 13, 13, 7, 17, 17, 7, 11, 11, 3, 11, 11, 5, 13, 13, 5, 7, 7, 1, 9, 9, 7, 15, 15, 7, 13, 13, 5, 21, 21, 11, 27, 27, 11, 17, 17, 5, 19, 19, 9, 23, 23, 9, 13, 13, 3, 19, 19, 13, 29, 29, 13, 23

Offset: 1

Author

Rémy Sigrist, Nov 22 2018

Keywords

• nonn
• tabf

Comments

This sequence and A321784 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

Examples

			The first rows are:
   1
   1, 3, 3
   1, 5, 5, 3, 7, 7, 3, 5, 5
		

Links

• Rémy Sigrist, Rows n = 1..9, flattened
• Index entries related to Pythagorean Triples

Crossrefs

Cf. A001333, A321768, A321769, A321784.

Programs

PARI

M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (sqrtint(t[3, 1] - t[2, 1]))

Formula

Empirically:

- T(n, 1) = 1,

- T(n, (3^(n-1) + 1)/2) = A001333(n),

- T(n, 3^(n-1)) = 2*n - 1.