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%I #12 Jan 18 2023 03:28:43
%S 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,5,1,1,1,5,3,3,5,1,
%T 1,1,3,3,3,1,1,1,1,5,3,3,5,1,1,1,3,1,1,5,1,1,3,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,3,1,1,1,1,1,1,1,5,1,1,1,5,3,3,5,1
%N A family of triangles T(m), m >= 0, read by triangles and then by rows; triangle T(0) is [1; 1, 1]; for m >= 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; t, t; u, t+u+v, v; u, u, v, v].
%C We apply the following substitutions to transform T(m) into T(m+1):
%C t
%C / \
%C / \
%C t t-----t
%C / \ ___\ / \ / \
%C / \ / / \ / \
%C u-----v u---t+u+v---v
%C / \ / \ / \
%C / \ / \ / \
%C u-----u-----v-----v
%C and:
%C u-----u-----v-----v
%C \ / \ / \ /
%C \ / \ / \ /
%C u-----v u---t+u+v---v
%C \ / ___\ \ / \ /
%C \ / / \ / \ /
%C t t-----t
%C \ /
%C \ /
%C t
%C T(m) has 3^m+1 rows.
%C All terms are odd.
%C As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).
%H Rémy Sigrist, <a href="/A356097/a356097.png">Colored representation of T(6)</a> (the color is function of T(6)(n,k))
%H Rémy Sigrist, <a href="/A356097/a356097_1.png">Representation of the multiples of 3 in T(7)</a>
%H Rémy Sigrist, <a href="/A356097/a356097_2.png">Representation of the multiples of 5 in T(7)</a>
%H Rémy Sigrist, <a href="/A356097/a356097_3.png">Representation of the multiples of 7 in T(7)</a>
%H Rémy Sigrist, <a href="/A356097/a356097_4.png">Representation of the 1's in T(7)</a>
%H Rémy Sigrist, <a href="/A356097/a356097_5.png">Representation of the terms congruent to 1 mod 4 in T(7)</a>
%H Rémy Sigrist, <a href="/A356097/a356097.gp.txt">PARI program</a>
%H Rémy Sigrist, <a href="https://arxiv.org/abs/2301.06039">Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp</a>, arXiv:2301.06039 [math.CO], 2023.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/N-flake#Hexaflake">Hexaflake</a>
%e Triangle T(0) is:
%e 1
%e 1 1
%e Triangle T(1) is:
%e 1
%e 1 1
%e 1 3 1
%e 1 1 1 1
%e Triangle T(2) is:
%e 1
%e 1 1
%e 1 3 1
%e 1 1 1 1
%e 1 1 5 1 1
%e 1 5 3 3 5 1
%e 1 1 3 3 3 1 1
%e 1 1 5 3 3 5 1 1
%e 1 3 1 1 5 1 1 3 1
%e 1 1 1 1 1 1 1 1 1 1
%o (PARI) See Links section.
%Y See A355855, A356002, A356096 and A356098 for similar sequences.
%Y Cf. A353174.
%K nonn,tabf
%O 0,8
%A _Rémy Sigrist_, Jul 26 2022
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