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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. 5

%I #16 Aug 09 2017 20:18:01

%S 2,2,3,4,5,7,9,12,16,21,28,37,49,67,91

%N 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 a(4)=1. A dissection into 5 triangles is impossible.

%C 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 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 a(11)=7:

%e *

%e / \

%e / \

%e / \

%e / \

%e / \

%e / \

%e / \

%e / \

%e / \

%e / 7 \

%e / \

%e / \

%e / \

%e *-----------*---------------*

%e / \ / \ / \

%e / \ 3 / \ / \

%e / 2 \ / \ 4 / \

%e *-------* / \ / \

%e / \ 2 / \ / 4 \ / 4 \

%e / \ *---* \ / \

%e / 2 \ / \ / \ / \

%e *-------*---*---------------*---------------*

%e More illustrations are provided on pages 17-19 of the Drapal and Hamalainen article.

%Y Cf. A167123, A290653.

%K nonn,hard,more

%O 6,1

%A _Hugo Pfoertner_, Aug 09 2017

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Last modified April 23 10:29 EDT 2024. Contains 371905 sequences. (Running on oeis4.)