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

Author

Rémy Sigrist, Aug 17 2022

Keywords

• nonn

Comments

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

Examples

			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}
		

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.

Crossrefs

Cf. A329919, A329927.

Programs

PARI

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

Formula

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