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A355855 A family of triangles T(m), m > 0, read by triangles and then by rows; triangle T(1) 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+u, t+v; u, u+v, v].

%I A355855 #24 Jan 18 2023 14:52:14
%S A355855 1,1,1,1,2,2,1,2,1,1,3,3,2,4,2,3,4,4,3,1,3,2,3,1,1,4,4,3,6,3,5,7,7,5,
%T A355855 2,6,4,6,2,5,6,8,8,6,5,3,7,4,8,4,7,3,4,6,7,6,6,7,6,4,1,4,3,5,2,5,3,4,
%U A355855 1,1,5,5,4,8,4,7,10,10,7,3,9,6,9,3,8,10,13,13,10,8
%N A355855 A family of triangles T(m), m > 0, read by triangles and then by rows; triangle T(1) 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+u, t+v; u, u+v, v].
%C A355855 We apply the following substitutions to transform T(m) into T(m+1):
%C A355855                            t
%C A355855                           / \
%C A355855         t                /   \
%C A355855        / \     -->     t+u---t+v
%C A355855       u---v            / \   / \
%C A355855                       /   \ /   \
%C A355855                      u----u+v----v
%C A355855 This sequence can be seen as a two-dimensional variant of A049456.
%C A355855 The base of T(m) corresponds to the m-th row of A049456.
%C A355855 T(m) has 2^(m-1)+1 rows, and largest term 2^(m-1).
%C A355855 As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).
%H A355855 Rémy Sigrist, <a href="/A355855/b355855.txt">Table of n, a(n) for n = 1..11313</a>
%H A355855 Rémy Sigrist, <a href="/A355855/a355855.png">Representation of multiples of 2 in T(12)</a>
%H A355855 Rémy Sigrist, <a href="/A355855/a355855_4.png">Representation of multiples of 2^2 in T(12)</a>
%H A355855 Rémy Sigrist, <a href="/A355855/a355855_1.png">Representation of multiples of 3 in T(12)</a>
%H A355855 Rémy Sigrist, <a href="/A355855/a355855_2.png">Representation of multiples of 5 in T(12)</a>
%H A355855 Rémy Sigrist, <a href="/A355855/a355855_3.png">Colored representation of T(10)</a> (the color is function of T(10)(n,k))
%H A355855 Rémy Sigrist, <a href="/A355855/a355855.gp.txt">PARI program</a>
%H A355855 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.
%e A355855 T(1) is:
%e A355855           1
%e A355855          1 1
%e A355855 T(2) is:
%e A355855           1
%e A355855          2 2
%e A355855         1 2 1
%e A355855 T(3) is:
%e A355855           1
%e A355855          3 3
%e A355855         2 4 2
%e A355855        3 4 4 3
%e A355855       1 3 2 3 1
%e A355855 T(4) is:
%e A355855           1
%e A355855          4 4
%e A355855         3 6 3
%e A355855        5 7 7 5
%e A355855       2 6 4 6 2
%e A355855      5 6 8 8 6 5
%e A355855     3 7 4 8 4 7 3
%e A355855    4 6 7 6 6 7 6 4
%e A355855   1 4 3 5 2 5 3 4 1
%o A355855 (PARI) See Links section.
%o A355855 (PARI) T(m,n,k) = { if (m==1, 1, my (nn=(n+1)\2, kk=(k+1)\2); if (n%2==1 && k%2==1, T(m-1, nn, kk), n%2==1 && k%2==0, T(m-1, nn, kk) + T(m-1, nn, kk+1), n%2==0 && k%2==1, T(m-1, nn, kk) + T(m-1, nn+1, kk), T(m-1, nn, kk) + T(m-1, nn+1, kk+1))) }
%Y A355855 Cf. A049456.
%K A355855 nonn,easy,tabf
%O A355855 1,5
%A A355855 _Rémy Sigrist_, Jul 19 2022