A134588 A modified Heron sequence starting from 1, 2.

1, 2, 3, 10, 27, 98, 120, 327, 196745, 277248, 312987, 405769, 456300, 575532, 702219

Offset

1,2

Comments

A modified Heron sequence is an increasing sequence such that every three consecutive terms (say u, v, w) of which determine a Heron triangle by using u+v, u+w and v+w as three sizes. A Heron triangle is a triangle with integer sides and integer area.

Given u<v be positive integers such that (u, v) is not (1, 4), (1, 9), (2, 8), (2, 18) or (4, 6). Then there is an integer w such that the three sizes u+v, u+w and v+w form a Heron triangle. Therefore infinite modified Heron sequence exists. We can construct arbitrarily long Heron sequences. However, it is still open whether an infinite Heron sequence exists.

Links

Table of n, a(n) for n=1..15.

Paul Yiu, K. R. S. Sastry and Shanzhen Gao, Heron Sequences, presented on the 2007 Integers Conference and INTEGERS - Electronic Journal of Combinatorial Number Theory, Article 16 Volume 9 Supplement (2009).

Crossrefs

Cf. A134587.

Sequence in context: A278088 A052929 A151415 * A000060 A089752 A264759

Adjacent sequences: A134585 A134586 A134587 * A134589 A134590 A134591

Keyword

nonn,more

Author

Shanzhen Gao, Nov 02 2007

Extensions

a(9)-a(15) from Rémy Sigrist, Jun 03 2019

Status

approved