A390230 Norms of primitive Pythagorean quadruples (PPQs) derived from primitive Pythagorean triangles (PPTs) ordered within a Berggren ternary tree of PPTs.

7, 19, 39, 21, 37, 97, 67, 129, 229, 103, 93, 133, 43, 61, 181, 139, 337, 577, 247, 277, 417, 147, 273, 673, 439, 733, 1333, 619, 481, 661, 199, 219, 499, 301, 403, 763, 373, 223, 283, 73, 91, 291, 237, 643, 1083, 453, 559, 859, 313, 727, 1767, 1137, 1831, 3351

Offset

1,1

Comments

A Berggren ternary tree is an infinite ordering of all PPTs with (3,4,5) as root node. Let the hypotenuse, odd leg, even leg and semiperimeter of a PPT be h, o, e and s respectively. Then, if a = e/2, b = s-o, c = s and d = h+e/2, it gives a^2 + b^2 + c^2 = d^2 as a PPQ. Such derived PPQs are a subsequence of the sequence of all PPQs. The Berggren tree ordering of PPTs is as follows:

(3)

(4)

(5)

__________/ | \__________

/ | \

( 5) (21) (15)

(12) (20) ( 8)

(13) (29) (17)

/ | \ / | \ / | \

( 7) (55) (45) (39) (119) (77) (33) (65) (35)

(24) (48) (28) (80) (120) (36) (56) (72) (12)

(25) (73) (53) (89) (169) (85) (65) (97) (37)

/|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\

* * ** * ** * * * * ** * ** * * * * ** * ** * *

Links

Table of n, a(n) for n=1..54.

B. Berggren, Pytagoreiska Trianglar, Tidskrift för elementär matematik, fysik och kemi 17 (1934), 129-139 (English translation).

Frank M Jackson, Mathematica program.

P. Oliverio, Self-Generating Pythagorean Quadruples and N-tuples.", Fib. Quart. 34, 98-101, 1996.

Wikipedia, Tree of primitive Pythagorean triples.

Wikipedia, Pythagorean quadruple.

Example

The sequence of norms of PPQs derived from PPTs that aligns with the Berggren ternary tree is as follows:-
                         7
           __________/   |   \__________
          /              |              \
        19              39               21
    /   |   \        /   |   \        /   |   \
   37  97   67     129  229  103    93   133  43
  /|\  /|\  /|\    /|\  /|\  /|\    /|\  /|\  /|\
 * * ** * ** * *  * * ** * ** * *  * * ** * ** * *
a(13) = 43, where 43 is the norm of PPQ derived from the PPT (35,12,37). The PPQ is (6,7,42,43).

Mathematica

(* See link above *)

Crossrefs

Cf. A096910, A097263.

Sequence in context: A051937 A119327 A152728 * A252789 A099061 A078163

Adjacent sequences: A390227 A390228 A390229 * A390231 A390232 A390233

Keyword

nonn,tabf

Author

Frank M Jackson, Feb 25 2026

Status

approved