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%I #17 Jul 07 2023 14:51:30
%S 1,1,1,2,2,3,5,7,12,22,37,66,118,228,413,762,1441,2718,5147,9804,
%T 18594,35420,67729,129976,249176,479112,921625,1777474,3429822,
%U 6632522,12826031,24850926,48165224,93507918,181566683,353075420,686711066,1337373564,2604998105,5080131368
%N The number of equal-sized equilateral triangles in the highest stack of triangles contained in successive Genealodrons formed from 2^n - 1 same size equilateral triangles
%C As explained in the comments to A179178 the n-th Genealodron can be formed by adding 2^(n-1) same size equilateral triangles to the left and right edges of the last 2^(n-2) triangles added to the (n-1)th Genealodron. It is easier however to imagine the n-th Genealodron formed by taking a new same size equilateral triangle and joining the bottom edge of the first triangle of a (n-1)th Genealodron to its left edge and similarly the bottom edge of the first triangle of another (n-1)th Genealodron to its right edge.
%C The shape formed is the same. Expressed in genealogical terms, instead of adding a round of equivalent many-times great-grandparent triangles to the structure, a child triangle has been put in forcing each triangle to move up a generation on both the father and mother's side.
%C The overlaps become increasingly complex as the equilateral triangles stack into spirals within the structure and as n gets larger the child triangle method becomes the only feasible way of generating successive Genealodrons.
%C For n>=18, with reference to the illustration of the initial terms of A179178, the location of the highest stack of triangles will stabilize at the cell labeled 11 for even n and at the cells labeled 5 and 6 for odd n. The sequence can be computed as the number of walks in the honeycomb lattice of length less than or equal to n that don't double back on themselves and that start at the origin and finish at the location with the greatest number of such walks. Also when making the first step only two of the three cells adjacent to the origin must be considered. - _Andrew Howroyd_, Mar 26 2016
%H Andrew Howroyd, <a href="/A179316/b179316.txt">Table of n, a(n) for n = 1..250</a>
%H Kival Ngaokrajang, <a href="/A179178/a179178.pdf">Illustration of initial terms</a>, A179178.
%e From _Andrew Howroyd_, Mar 26 2016: (Start)
%e See illustration of initial terms of A179178 for cell numbering.
%e a(4) = 2 because there are 2 permissible walks of length <= 4 ending on the cell labeled 11: {1,3,6,11} and {1,2,5,11}.
%e a(6) = 3 because there are 3 permissible walks of length <= 6 ending on the cell labeled 10: {1,2,5,10}, {1,3,6,11,5,10} and {1,2,4,9,18,10}.
%e (End)
%Y A179178 is a related Genealodron sequence.
%K nonn
%O 1,4
%A Elizabeth Hignell (elizabethhignell(AT)hotmail.com), Jul 10 2010
%E a(13)-a(40) from _Andrew Howroyd_, Mar 25 2016
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