A329919 -id:A329919 - cp's OEIS frontend

A329927 a(n) is the number of squares with largest size after n iterations of the "Square Multiscale" substitution.

1, 1, 1, 1, 16, 1, 32, 1, 48, 1, 64, 256, 1, 80, 768, 1, 96, 1536, 1, 112, 2560, 4096, 1, 128, 3840, 16384, 1, 144, 5376, 40960, 1, 160, 7168, 81920, 65536, 1, 176, 9216, 143360, 327680, 1, 192, 11520, 229376, 983040, 1, 208, 14080, 344064, 2293760, 1048576, 1

Offset: 0

Rémy Sigrist, Nov 24 2019

nonn

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

This sequence only contains ones and multiples of 16; the ones correspond to situations where the central square is the largest.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A329927
Yotam Smilansky, Yaar Solomon, Multiscale Substitution Tilings, arXiv:2003.11735 [math.DS], 2020.

Cf. A328074, A329919.

PARI
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a(n) = (A329919(n+1) - A329919(n)) / 16.

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

Original entry on oeis.org

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

Rémy Sigrist, Aug 17 2022

nonn

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

			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}

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.

Cf. A329919, A329927.

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

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

A356624 After n iterations of the "Square Multiscale" substitution, the largest tiles have side length 3^t / 5^f; a(n) = t (A356625 gives corresponding f's).

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3, 0, 13, 10, 7, 4, 1, 14, 11, 8, 5, 2, 15, 12, 9, 6, 3, 0, 16, 13, 10, 7, 4, 1, 17, 14, 11, 8, 5, 2, 18, 15, 12, 9, 6, 3, 0, 19, 16, 13, 10, 7, 4, 1, 20, 17

Offset: 0

Rémy Sigrist, Aug 17 2022

nonn

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

			The first terms, alongside the corresponding side lengths, are:
  n   a(n)  Side length
  --  ----  -----------
   0     0            1
   1     1          3/5
   2     2         9/25
   3     3       27/125
   4     0          1/5
   5     4       81/625
   6     1         3/25
   7     5     243/3125
   8     2        9/125
   9     6    729/15625
  10     3       27/625

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.

Cf. A022336, A329919, A354535, A356625.

{ sc = [1]; for (n=0, 78, s = vecmax(sc); print1 (valuation(s,3)", "); sc = setunion(setminus(sc,[s]), Set([3*s/5, s/5]))) }

5^A356625(n) >= 3^a(n).

A356625 After n iterations of the "Square Multiscale" substitution, the largest tiles have side length 3^t / 5^f; a(n) = f (A356624 gives corresponding t's).

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 2, 5, 3, 6, 4, 2, 7, 5, 3, 8, 6, 4, 9, 7, 5, 3, 10, 8, 6, 4, 11, 9, 7, 5, 12, 10, 8, 6, 4, 13, 11, 9, 7, 5, 14, 12, 10, 8, 6, 15, 13, 11, 9, 7, 5, 16, 14, 12, 10, 8, 6, 17, 15, 13, 11, 9, 7, 18, 16, 14, 12, 10, 8, 6, 19, 17, 15, 13, 11, 9, 7

Offset: 0

Rémy Sigrist, Aug 17 2022

nonn

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

			The first terms, alongside the corresponding side lengths, are:
  n   a(n)  Side length
  --  ----  -----------
   0     0            1
   1     1          3/5
   2     2         9/25
   3     3       27/125
   4     1          1/5
   5     4       81/625
   6     2         3/25
   7     5     243/3125
   8     3        9/125
   9     6    729/15625
  10     4       27/625

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.

Cf. A022337, A329919, A354535, A356624.

{ sc = [1]; for (n=0, 76, s = vecmax(sc); print1 (-valuation(s,5)", "); sc = setunion(setminus(sc,[s]), Set([3*s/5, s/5]))) }

5^a(n) >= 3^A356624(n).

cp's OEIS Frontend

A329927 a(n) is the number of squares with largest size after n iterations of the "Square Multiscale" substitution.

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

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Author

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Crossrefs

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PARI

Formula

A356624 After n iterations of the "Square Multiscale" substitution, the largest tiles have side length 3^t / 5^f; a(n) = t (A356625 gives corresponding f's).

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Examples

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Crossrefs

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PARI

Formula

A356625 After n iterations of the "Square Multiscale" substitution, the largest tiles have side length 3^t / 5^f; a(n) = f (A356624 gives corresponding t's).

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Examples

Links

Crossrefs

Programs

PARI

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