|
|
A134588
|
|
A modified Heron sequence starting from 1, 2.
|
|
2
|
|
|
1, 2, 3, 10, 27, 98, 120, 327, 196745, 277248, 312987, 405769, 456300, 575532, 702219
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
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
|
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|