A321783 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321770(n, k) - A321768(n, k))/2).

1, 2, 2, 1, 3, 3, 2, 5, 5, 2, 4, 4, 1, 4, 4, 3, 8, 8, 3, 7, 7, 2, 8, 8, 5, 12, 12, 5, 9, 9, 2, 7, 7, 4, 9, 9, 4, 6, 6, 1, 5, 5, 4, 11, 11, 4, 10, 10, 3, 13, 13, 8, 19, 19, 8, 14, 14, 3, 12, 12, 7, 16, 16, 7, 11, 11, 2, 11, 11, 8, 21, 21, 8, 18, 18, 5, 19, 19

Offset: 1

Author

Rémy Sigrist, Nov 19 2018

Keywords

• nonn
• tabf

Comments

This sequence and A321782 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

Examples

			The first rows are:
   1
   2, 2, 1
   3, 3, 2, 5, 5, 2, 4, 4, 1
		

Links

• Rémy Sigrist, Rows n = 1..9, flattened
• Kevin Ryde, Trees of Primitive Pythagorean Triples, see section UAD Tree, "row-wise q".
• Robert Saunders and Trevor Randall, The Family Tree of the Pythagorean Triplets Revisited, Mathematical Gazette, item 78.12, volume 78, July 1994, pages 190-193, see page 192 tree terms "n" by columns.
• Index entries related to Pythagorean Triples

Crossrefs

Cf. A000129, A321768, A321770, A321782.

Cf. A001653 (row sums).

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[1, 1])/2))

Formula

Empirically:

- T(n, 1) = n,

- T(n, (3^(n-1) + 1)/2) = A000129(n),

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