A321782 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = sqrt((A321768(n, k) + A321770(n, k))/2).

2, 3, 5, 4, 4, 8, 7, 8, 12, 9, 7, 9, 6, 5, 11, 10, 13, 19, 14, 12, 16, 11, 11, 21, 18, 19, 29, 22, 16, 20, 13, 10, 18, 15, 14, 22, 17, 11, 13, 8, 6, 14, 13, 18, 26, 19, 17, 23, 16, 18, 34, 29, 30, 46, 35, 25, 31, 20, 17, 31, 26, 25, 39, 30, 20, 24, 15, 14, 30

Offset: 1

Author

Rémy Sigrist, Nov 18 2018

Keywords

• nonn
• tabf

Comments

This sequence and A321783 are related to a parametrization of the primitive Pythagorean triples in the tree described in A321768.

This sequence is "i" from the construction in A321768. It takes ternary digits of k-1 from most to least significant. Here the result is the same going instead least to most, due to how the relevant matrix product is related to its reversal. As a flat sequence this means a(A351702(n)) = a(n) unchanged. - Kevin Ryde, Mar 10 2022

Examples

			The first rows are:
   2
   3, 5, 4
   4, 8, 7, 8, 12, 9, 7, 9, 6
		

Links

• Rémy Sigrist, Rows n = 1..9, flattened
• Kevin Ryde, Trees of Primitive Pythagorean Triples, section UAD Tree, "row-wise p".
• 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 "m" by columns.
• Index entries related to Pythagorean Triples

Crossrefs

Cf. A000129, A321768, A321770, A321783.

Cf. A001542 (row sums).

Cf. A351702 (product reversal permutation).

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[1, 1] + t[3, 1])/2))

Formula

Empirically:

- T(n, 1) = n + 1,

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

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