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

1,1

Comments

The tree P runs uniquely through every primitive Pythagorean triple.

See A321768 for additional comments about P.

All terms are odd.

Links

Rémy Sigrist, Rows n = 1..9, flattened

Wikipedia, Tree of primitive Pythagorean triples

Index entries related to Pythagorean Triples

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).

Example

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

Prog

(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])

Crossrefs

See A321768 and A321769 for the other components.

Cf. A001653, A001844, A053755.

Sequence in context: A189485 A226616 A226618 * A322926 A178854 A224339

Adjacent sequences: A321767 A321768 A321769 * A321771 A321772 A321773

Keyword

nonn,tabf

Author

Rémy Sigrist, Nov 18 2018

Status

approved