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.

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

Offset: 6

Author

Hugo Pfoertner, Aug 09 2017

Keywords

• nonn
• hard
• more

Comments

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

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.

Examples

			a(11)=7:
                        *
                       / \
                      /   \
                     /     \
                    /       \
                   /         \
                  /           \
                 /             \
                /               \
               /                 \
              /         7         \
             /                     \
            /                       \
           /                         \
          *-----------*---------------*
         / \         / \             / \
        /   \   3   /   \           /   \
       /  2  \     /     \    4    /     \
      *-------*   /       \       /       \
     / \  2  / \ /    4    \     /    4    \
    /   \   *---*           \   /           \
   /  2  \ / \ /             \ /             \
  *-------*---*---------------*---------------*
More illustrations are provided on pages 17-19 of the Drapal and Hamalainen article.
		

Links

• Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.

Crossrefs

Cf. A167123, A290653.