A356002 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; t+2*u, t+u+v, t+2*v; u, 2*u+v, u+2*v, v].

1, 1, 1, 1, 3, 3, 3, 3, 3, 1, 3, 3, 1, 1, 5, 5, 7, 7, 7, 3, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 3, 9, 9, 3, 7, 9, 9, 9, 9, 9, 9, 7, 5, 7, 9, 9, 9, 9, 9, 7, 5, 1, 5, 7, 3, 9, 9, 3, 7, 5, 1, 1, 7, 7, 11, 11, 11, 5, 15, 15, 5, 17, 17, 17, 17, 17

Offset

0,5

Comments

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

t

/ \

/ \

t 2*t+u 2*t+v

/ \ ___\ / \ / \

/ \ / / \ / \

u-----v t+2*u t+u+v t+2*v

/ \ / \ / \

/ \ / \ / \

u---2*u+v--u+2*v--v

and:

u---2*u+v--u+2*v--v

\ / \ / \ /

\ / \ / \ /

u-----v t+2*u t+u+v t+2*v

\ / ___\ \ / \ /

\ / / \ / \ /

t 2*t+u 2*t+v

\ /

\ /

t

T(m) has 3^m+1 rows, and largest term 3^m.

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..82.

Rémy Sigrist, Colored representation of T(6) (the color is function of T(6)(n,k))

Rémy Sigrist, Colored representation of T(6) (the color is function of the 3-adic valuation of T(6)(n,k))

Rémy Sigrist, Representation of the terms congruent to 3 mod 4 in T(6)

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

Triangle T(0) is:
              1
             1 1
Triangle T(1) is:
              1
             3 3
            3 3 3
           1 3 3 1
Triangle T(2) is:
              1
             5 5
            7 7 7
           3 9 9 3
          9 9 9 9 9
         9 9 9 9 9 9
        3 9 9 3 9 9 3
       7 9 9 9 9 9 9 7
      5 7 9 9 9 9 9 7 5
     1 5 7 3 9 9 3 7 5 1

Prog

(PARI) See Links section.

Crossrefs

See A355855 for a similar sequence.

Cf. A177407.

Sequence in context: A247655 A097675 A141605 * A251551 A073139 A122845

Adjacent sequences: A355999 A356000 A356001 * A356003 A356004 A356005

Keyword

nonn,tabf

Author

Rémy Sigrist, Jul 22 2022

Status

approved