A356096 - OEIS

%I #11 Jan 18 2023 03:28:47

%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,-1,1,1,5,5,5,5,

%T 1,1,-1,5,3,5,-1,1,1,1,5,5,5,5,1,1,1,3,1,-1,5,-1,1,3,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,3,1,1,1,1,1,1,-1,5,-1,1

%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; 2*t-u, 2*t-v; 2*u-t, t+u+v, 2*v-t; u, 2*u-v, 2*v-u, v].

%C We apply the following substitutions to transform T(m) into T(m+1):

%C                             t

%C                            / \

%C                           /   \

%C        t               2*t-u 2*t-v

%C       / \    ___\       / \   / \

%C      /   \      /      /   \ /   \

%C     u-----v         2*u-t t+u+v 2*v-t

%C                      / \   / \   / \

%C                     /   \ /   \ /   \

%C                    u---2*u-v--2*v-u--v

%C and:

%C                    u---2*u-v--2*v-u--v

%C                     \   / \   / \   /

%C                      \ /   \ /   \ /

%C     u-----v         2*u-t t+u+v 2*v-t

%C      \   /   ___\      \   / \   /

%C       \ /       /       \ /   \ /

%C        t               2*t-u 2*t-v

%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="/A356096/a356096.png">Colored representation of T6</a> (the color is function of T(6)(n, k))

%H Rémy Sigrist, <a href="/A356096/a356096_1.png">Representation of the multiples of 3 in T(7)</a>

%H Rémy Sigrist, <a href="/A356096/a356096_2.png">Representation of the negative terms in T(7)</a>

%H Rémy Sigrist, <a href="/A356096/a356096.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.

%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   5   5   5   1

%e             1  -1   5   3   5  -1   1

%e           1   1   5   5   5   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, A356097 and A356098 for similar sequences.

%K sign,tabf

%O 0,8

%A _Rémy Sigrist_, Jul 26 2022