The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #42 Mar 05 2026 12:36:43
%S 7,19,39,21,37,97,67,129,229,103,93,133,43,61,181,139,337,577,247,277,
%T 417,147,273,673,439,733,1333,619,481,661,199,219,499,301,403,763,373,
%U 223,283,73,91,291,237,643,1083,453,559,859,313,727,1767,1137,1831,3351
%N Norms of primitive Pythagorean quadruples (PPQs) derived from primitive Pythagorean triangles (PPTs) ordered within a Berggren ternary tree of PPTs.
%C 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:
%C (3)
%C (4)
%C (5)
%C __________/ | \__________
%C / | \
%C ( 5) (21) (15)
%C (12) (20) ( 8)
%C (13) (29) (17)
%C / | \ / | \ / | \
%C ( 7) (55) (45) (39) (119) (77) (33) (65) (35)
%C (24) (48) (28) (80) (120) (36) (56) (72) (12)
%C (25) (73) (53) (89) (169) (85) (65) (97) (37)
%C /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\
%C * * ** * ** * * * * ** * ** * * * * ** * ** * *
%H B. Berggren, <a href="https://www.researchgate.net/publication/326058164">Pytagoreiska Trianglar</a>, Tidskrift för elementär matematik, fysik och kemi 17 (1934), 129-139 (English translation).
%H Frank M Jackson, <a href="/A390230/a390230.txt">Mathematica program</a>.
%H P. Oliverio, <a href="https://www.fq.math.ca/Scanned/34-2/oliverio.pdf">Self-Generating Pythagorean Quadruples and N-tuples."</a>, Fib. Quart. 34, 98-101, 1996.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples">Tree of primitive Pythagorean triples</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pythagorean_quadruple">Pythagorean quadruple</a>.
%e The sequence of norms of PPQs derived from PPTs that aligns with the Berggren ternary tree is as follows:-
%e 7
%e __________/ | \__________
%e / | \
%e 19 39 21
%e / | \ / | \ / | \
%e 37 97 67 129 229 103 93 133 43
%e /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\
%e * * ** * ** * * * * ** * ** * * * * ** * ** * *
%e a(13) = 43, where 43 is the norm of PPQ derived from the PPT (35,12,37). The PPQ is (6,7,42,43).
%t (* See link above *)
%Y Cf. A096910, A097263.
%K nonn,tabf
%O 1,1
%A _Frank M Jackson_, Feb 25 2026