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

Author

Rémy Sigrist, Jul 19 2022

Keywords

• nonn
• easy
• tabf

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).

Examples

			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
		

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.

Crossrefs

Cf. A049456.

Programs

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))) }