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A290821 Side length of largest equilateral triangle that can be made from n or fewer equilateral triangles with integer sides s_k, subject to gcd(s_1,s_2,...,s_n) = 1.

%I A290821 #34 Nov 23 2024 08:29:20
%S A290821 1,1,1,2,2,3,4,5,7,9,12,16,21,28,39,49
%N A290821 Side length of largest equilateral triangle that can be made from n or fewer equilateral triangles with integer sides s_k, subject to gcd(s_1,s_2,...,s_n) = 1.
%C A290821 No construction from 2, 3 or 5 equilateral triangles exists. The first difference from the Padovan numbers occurs for a(15)=39, where the corresponding term A000931(19)=37. a(16)=A000931(20)=49. a(n) >= A000931(n+3). From the growth behavior of A290697 it is conjectured that a(k) > A000931(k+3) for all k > 20.
%C A290821 a(19) is at least 130. This compares with A000931(23) = 114. It hints of growth behavior similar to sqrt(A014529) or sqrt(A001590). Ceiling(sqrt(A001590(n))) matches a(n) to n=14, then runs 38, 52, 70, 95, 128, ... . - _Peter Munn_, Mar 10 2018
%C A290821 From _Peter Munn_, Mar 14 2018 re monotonicity: (Start)
%C A290821 For n >= 6, a(n+1) > a(n).
%C A290821 Sketch of proof (inductive step) expressed in terms of tiling:
%C A290821 Given a triangle of side a(n) tiled with n equilateral triangular tiles. Let X, Y and Z be the tiles incident on its vertices, with X being not smaller than Y or Z.
%C A290821 Case 1: Y and Z have no vertices coincident. Remove Y and Z, thereby reducing the tiled area to a pentagon that has edges A and C that were previously internal to the area, and an edge B between A and C. Fit a new tile T against edge B, thereby extending edges A and C. Make the tiled area triangular by fitting a new tile against each of the extended edges.
%C A290821 Case 2: X, Y and Z have pairwise coincident vertices. It follows that these tiles are the same size. Remove Y and Z, thereby reducing the tiled area to a rhombus. Remove the tile at the rhombus vertex opposite X. The remaining area is a pentagon, since n >= 6. Extend the area by resiting Y against X, and Z against Y so that X and Z have external edges aligned. Make the area trapezoidal by fitting a new tile against the area's edge that includes an edge of Y. Fit another tile T against the smaller of the trapezoid's parallel edges.
%C A290821 In each case, we now have n+1 tiles, tiling an equilateral triangle with side length a(n) plus the side of T. As the sides of new and removed tiles can be calculated by adding sides of tiles that stayed in place, the GCD of the sides is unchanged.
%C A290821 (End)
%H A290821 Stuart Anderson, <a href="http://web.archive.org/web/20170806090944/http://www.squaring.net/tri/tritri/tet.html">An Introduction to Triangled Equilateral Triangles</a>
%H A290821 Ales Drapal and Carlo Hamalainen, <a href="http://arxiv.org/abs/0910.5199">An enumeration of equilateral triangle dissections</a>, arXiv:0910.5199 [math.CO], 2009-2010.
%H A290821 Hugo Pfoertner, <a href="/A290821/a290821.pdf">Illustration of a(15)=39</a>.
%e A290821 a(12) = 16:
%e A290821                                   *
%e A290821                                  / \
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%e A290821                               +       +
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%e A290821                             +           +
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%e A290821                           +               +
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%e A290821                         +                   +
%e A290821                        /                     \
%e A290821                       +                       +
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%e A290821                     +                           +
%e A290821                    /                             \
%e A290821                   +                               +
%e A290821                  /                                 \
%e A290821                 *---+---*---+---+---+---+---+---+---*
%e A290821                / \     / \                         / \
%e A290821               +   +   +   +                       +   +
%e A290821              /     \ /     \                     /     \
%e A290821             *---*---*       +                   +       +
%e A290821            / \ / \ /         \                 /         \
%e A290821           +   *---*---+---+---*               +           +
%e A290821          /     \             / \             /             \
%e A290821         +       +           +   +           +               +
%e A290821        /         \         /     \         /                 \
%e A290821       +           +       +       +       +                   +
%e A290821      /             \     /         \     /                     \
%e A290821     +               +   +           +   +                       +
%e A290821    /                 \ /             \ /                         \
%e A290821   *---+---+---+---+---*---+---+---+---*---+---+---+---+---+---+---*
%Y A290821 Cf. A000931, A001590, A167123, A290653, A290697, A290820.
%Y A290821 A014529 gives greatest area of any convex polygon constructable from such triangles.
%Y A290821 A089047 is this sequence's equivalent for squares.
%K A290821 nonn,hard,more
%O A290821 1,4
%A A290821 _Hugo Pfoertner_, Aug 11 2017
%E A290821 Definition modified and 5 terms prepended by _Peter Munn_, Mar 14 2018