A356097 - 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].

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, 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, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 3, 3, 5, 1

Offset: 0

Rémy Sigrist, Jul 26 2022

We apply the following substitutions to transform T(m) into T(m+1):
                            t
                           / \
                          /   \
       t                 t-----t
      / \    _\       / \   / \
     /   \      /      /   \ /   \
    u-----v           u---t+u+v---v
                     / \   / \   / \
                    /   \ /   \ /   \
                   u-----u-----v-----v
and:
                   u-----u-----v-----v
                    \   / \   / \   /
                     \ /   \ /   \ /
    u-----v           u---t+u+v---v
     \   /   _\      \   / \   /
      \ /       /       \ /   \ /
       t                 t-----t
                          \   /
                           \ /
                            t
T(m) has 3^m+1 rows.
All terms are odd.
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).

			Triangle T(0) is:
              1
             1 1
Triangle T(1) is:
              1
             1 1
            1 3 1
           1 1 1 1
Triangle T(2) is:
              1
             1 1
            1 3 1
           1 1 1 1
          1 1 5 1 1
         1 5 3 3 5 1
        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

Rémy Sigrist, Colored representation of T(6) (the color is function of T(6)(n,k))
Rémy Sigrist, Representation of the multiples of 3 in T(7)
Rémy Sigrist, Representation of the multiples of 5 in T(7)
Rémy Sigrist, Representation of the multiples of 7 in T(7)
Rémy Sigrist, Representation of the 1's in T(7)
Rémy Sigrist, Representation of the terms congruent to 1 mod 4 in T(7)
Rémy Sigrist, PARI program
Rémy Sigrist, Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp, arXiv:2301.06039 [math.CO], 2023.
Wikipedia, Hexaflake

See A355855, A356002, A356096 and A356098 for similar sequences.

Cf. A353174.

PARI
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See Links section.
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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].

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