A354535 a(n) is the number of different tile sizes after n iterations of the "Square Multiscale" substitution.

1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20

Offset

0,2

Comments

See A329919 for further details about the "Square Multiscale" substitution.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Yotam Smilansky and Yaar Solomon, Multiscale Substitution Tilings, arXiv:2003.11735 [math.DS], 2020.

Formula

a(n+1) - a(n) = 0 or 1.

Example

The first terms, alongside the corresponding sizes, are:
  n  a(n)  Tile sizes
  -  ----  -----------------------------------------------
  0     1  {1}
  1     2  {3/5, 1/5}
  2     3  {9/25, 1/5, 3/25}
  3     4  {27/125, 1/5, 3/25, 9/125}
  4     5  {1/5, 81/625, 3/25, 9/125, 27/625}
  5     5  {81/625, 3/25, 9/125, 27/625, 1/25}
  6     6  {3/25, 243/3125, 9/125, 27/625, 1/25, 81/3125}
  7     6  {243/3125, 9/125, 27/625, 1/25, 81/3125, 3/125}

Prog

(PARI) { sc = [1]; for (n=0, 68, print1 (#sc", "); s = vecmax(sc); sc = setunion(setminus(sc, [s]), Set([3*s/5, s/5]))) }

Crossrefs

Cf. A329919, A329927.

Sequence in context: A061373 A327705 A104135 * A276656 A309250 A365339

Adjacent sequences: A354532 A354533 A354534 * A354536 A354537 A354538

Keyword

nonn

Author

Rémy Sigrist, Aug 17 2022

Status

approved