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

0,8

Comments

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).

Links

Table of n, a(n) for n=0..88.

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

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Cf. A353174.

Sequence in context: A030557 A086997 A074298 * A356096 A326029 A356167

Adjacent sequences: A356094 A356095 A356096 * A356098 A356099 A356100

Keyword

nonn,tabf

Author

Rémy Sigrist, Jul 26 2022

Status

approved