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].

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, 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, 1, 1, 5, 5, 4, 8, 4, 7, 10, 10, 7, 3, 9, 6, 9, 3, 8, 10, 13, 13, 10, 8

Offset

1,5

Comments

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

t

/ \

t / \

/ \ --> t+u---t+v

u---v / \ / \

/ \ / \

u----u+v----v

This sequence can be seen as a two-dimensional variant of A049456.

The base of T(m) corresponds to the m-th row of A049456.

T(m) has 2^(m-1)+1 rows, and largest term 2^(m-1).

As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..11313

Rémy Sigrist, Representation of multiples of 2 in T(12)

Rémy Sigrist, Representation of multiples of 2^2 in T(12)

Rémy Sigrist, Representation of multiples of 3 in T(12)

Rémy Sigrist, Representation of multiples of 5 in T(12)

Rémy Sigrist, Colored representation of T(10) (the color is function of T(10)(n,k))

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.

Example

T(1) is:
          1
         1 1
T(2) is:
          1
         2 2
        1 2 1
T(3) is:
          1
         3 3
        2 4 2
       3 4 4 3
      1 3 2 3 1
T(4) is:
          1
         4 4
        3 6 3
       5 7 7 5
      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 1

Prog

(PARI) See Links section.
(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))) }

Crossrefs

Cf. A049456.

Sequence in context: A180312 A178819 A369174 * A046816 A352248 A301475

Adjacent sequences: A355852 A355853 A355854 * A355856 A355857 A355858

Keyword

nonn,easy,tabf

Author

Rémy Sigrist, Jul 19 2022

Status

approved