A321770 Consider the tree of triples P(n, k) with n > 0 and 0 < k <= 3^(n-1), such that P(1, 1) = [3; 4; 5] and each triple t on some row branches to the triples A*t, B*t, C*t on the next row (with A = [1, -2, 2; 2, -1, 2; 2, -2, 3], B = [1, 2, 2; 2, 1, 2; 2, 2, 3] and C = [-1, 2, 2; -2, 1, 2; -2, 2, 3]); T(n, k) is the third component of P(n, k).

5, 13, 29, 17, 25, 73, 53, 89, 169, 85, 65, 97, 37, 41, 137, 109, 233, 425, 205, 193, 305, 125, 185, 505, 349, 505, 985, 509, 337, 481, 173, 149, 373, 241, 277, 565, 305, 157, 205, 65, 61, 221, 185, 445, 797, 377, 389, 629, 265, 493, 1325, 905, 1261, 2477

Offset: 1

Author

Rémy Sigrist, Nov 18 2018

Keywords

• nonn
• tabf

Comments

The tree P runs uniquely through every primitive Pythagorean triple.

See A321768 for additional comments about P.

All terms are odd.

Examples

			The first rows are:
   5
   13, 29, 17
   25, 73, 53, 89, 169, 85, 65, 97, 37
		

Links

• Rémy Sigrist, Rows n = 1..9, flattened
• Wikipedia, Tree of primitive Pythagorean triples
• Index entries related to Pythagorean Triples

Crossrefs

See A321768 and A321769 for the other components.

Cf. A001653, A001844, A053755.

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 (t[3, 1])

Formula

a(n)^2 = A321768(n)^2 + A321769(n)^2.

Empirically:

- T(n, 1) = A001844(n),

- T(n, (3^(n-1) + 1)/2) = A001653(n+1),

- T(n, 3^(n-1)) = A053755(n).