cp's OEIS Frontend

A356097 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].

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