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%I A134588 #15 Jun 03 2019 12:28:04
%S A134588 1,2,3,10,27,98,120,327,196745,277248,312987,405769,456300,575532,
%T A134588 702219
%N A134588 A modified Heron sequence starting from 1, 2.
%C A134588 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.
%C A134588 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.
%H A134588 Paul Yiu, K. R. S. Sastry and Shanzhen Gao, <a href="http://www.westga.edu/~integers/j16proc/j16proc.pdf">Heron Sequences</a>, presented on the 2007 Integers Conference and INTEGERS - Electronic Journal of Combinatorial Number Theory, Article 16 Volume 9 Supplement (2009).
%Y A134588 Cf. A134587.
%K A134588 nonn,more
%O A134588 1,2
%A A134588 _Shanzhen Gao_, Nov 02 2007
%E A134588 a(9)-a(15) from _Rémy Sigrist_, Jun 03 2019

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