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A290697 Size of largest triangle occurring in any of the possible dissections of an equilateral triangle into n equilateral triangles with integer sides, assuming gcd(s_1,s_2,...,s_n)=1, s_k being the side lengths.

%I A290697 #16 Aug 09 2017 20:18:01
%S A290697 2,2,3,4,5,7,9,12,16,21,28,37,49,67,91
%N A290697 Size of largest triangle occurring in any of the possible dissections of an equilateral triangle into n equilateral triangles with integer sides, assuming gcd(s_1,s_2,...,s_n)=1, s_k being the side lengths.
%C A290697 a(4)=1. A dissection into 5 triangles is impossible.
%C A290697 The size of the smallest triangle is 1 for triangles with maximum ratio of sizes between largest and smallest triangle for all n <= 20. If dissections with maximum size of largest occurring triangle and size of smallest triangle > 1 are found for larger n, there might be different configurations leading to a maximum ratio between largest and smallest side having a shorter largest side than the one provided as a(n). If this situation occurs for any n > 20, it shall be indicated in a corresponding comment.
%H A290697 Ales Drapal, Carlo Hamalainen, <a href="https://arxiv.org/abs/0910.5199">An enumeration of equilateral triangle dissections</a>, arXiv:0910.5199 [math.CO], 2009-2010.
%e A290697 a(11)=7:
%e A290697                         *
%e A290697                        / \
%e A290697                       /   \
%e A290697                      /     \
%e A290697                     /       \
%e A290697                    /         \
%e A290697                   /           \
%e A290697                  /             \
%e A290697                 /               \
%e A290697                /                 \
%e A290697               /         7         \
%e A290697              /                     \
%e A290697             /                       \
%e A290697            /                         \
%e A290697           *-----------*---------------*
%e A290697          / \         / \             / \
%e A290697         /   \   3   /   \           /   \
%e A290697        /  2  \     /     \    4    /     \
%e A290697       *-------*   /       \       /       \
%e A290697      / \  2  / \ /    4    \     /    4    \
%e A290697     /   \   *---*           \   /           \
%e A290697    /  2  \ / \ /             \ /             \
%e A290697   *-------*---*---------------*---------------*
%e A290697 More illustrations are provided on pages 17-19 of the Drapal and Hamalainen article.
%Y A290697 Cf. A167123, A290653.
%K A290697 nonn,hard,more
%O A290697 6,1
%A A290697 _Hugo Pfoertner_, Aug 09 2017