A346639 -id:A346639 - cp's OEIS frontend

A346640 Decimal expansion of 2 - log_3(2).

1, 3, 6, 9, 0, 7, 0, 2, 4, 6, 4, 2, 8, 5, 4, 2, 5, 6, 2, 9, 0, 0, 4, 7, 2, 8, 8, 5, 6, 5, 7, 2, 3, 9, 1, 4, 5, 7, 0, 0, 4, 1, 4, 3, 5, 9, 8, 6, 8, 1, 1, 9, 5, 7, 2, 1, 2, 9, 3, 4, 5, 0, 5, 6, 1, 6, 1, 3, 1, 4, 7, 9, 8, 6, 1, 9, 0, 8, 5, 1, 9, 4, 9, 3, 8, 8, 2, 7, 3, 1, 1, 4, 5, 0, 5, 4, 8, 2, 5, 4, 4, 3, 8, 6, 4

Offset: 1

Kevin Ryde, Jul 26 2021

Bedford and McMullen show this is the metric dimension of Hironaka's curve and equivalent carpets (see A346639).

			1.3690702464285425629004728856572391...

Timothy Bedford, Crinkly Curves, Markov Partitions and Dimension, Ph.D. thesis, University of Warwick, 1984, see proposition 4.1, page 89, cap(E) for the case s=2, t=2, r=3, Sum(k_i)=3 (and noting log(t) is a multiplier, not an exponent).
Curtis T. McMullen, Hausdorff Dimension of General Sierpinski Carpets, Nagoya Mathematical Journal, volume 96, number 19, 1984, pages 1-9, see page 2, m.dim(R) for the case m = s = 2 and n = r = 3.

Cf. A102525, A346639.

Cf. A213201.

RealDigits[2 - Log[3, 2], 10, 105][[1]] (* Amiram Eldar, Jul 27 2021 *)

2 - log(2)/log(3) \\ Michel Marcus, Jul 27 2021

Equals 2 - A102525.

Equals Sum_{d=1..2} d*log(1+1/d)/log(3). Compare with A213201. - Michel Marcus, Dec 25 2022

A343620 Decimal expansion of the Hausdorff dimension of 4 X 2 carpets with rows of 3 and 1 sub-parts.

Original entry on oeis.org

1, 4, 4, 9, 9, 8, 4, 3, 1, 3, 4, 7, 6, 4, 9, 5, 8, 4, 8, 9, 2, 1, 1, 6, 2, 5, 6, 0, 0, 6, 2, 3, 7, 9, 1, 5, 6, 6, 3, 5, 7, 3, 4, 7, 8, 4, 1, 2, 3, 3, 8, 1, 8, 9, 2, 5, 3, 1, 5, 4, 1, 2, 5, 8, 6, 3, 1, 1, 4, 6, 7, 4, 6, 9, 6, 3, 2, 5, 8, 6, 0, 6, 4, 1, 7, 9, 1, 8, 9, 9, 8, 4, 3, 0, 9, 1, 3, 9, 9, 3, 8, 5, 8, 8, 0

Offset: 1

Kevin Ryde, Aug 04 2021

Bedford (page 100 figure 34) gives this type of carpet as an example where the Hausdorff dimension differs from the capacity dimension (which is 3/2).
  +---+---+---+---+    Fractal carpet with each S
  |   | S | S | S |    a shrunken copy of the whole.
  +---+---+---+---+    Any 3 parts in one row and
  | S |   |   |   |    1 part in the other row.
  +---+---+---+---+

			1.4499843134764958489211625600623791...

Timothy Bedford, Crinkly Curves, Markov Partitions and Dimension, Ph.D. thesis, University of Warwick, 1984, chapter 4.
Curtis T. McMullen, Hausdorff Dimension of General Sierpinski Carpets, Nagoya Mathematical Journal, volume 96, number 19, 1984, pages 1-9, see page 1 dim(R) for the case n=4, m=2, t_0 = 1, t_1 = 3.

Cf. A346639 (3 X 2 carpets), A090388 (1+sqrt(3)).

RealDigits[Log2[1 + Sqrt[3]], 10, 100][[1]] (* Amiram Eldar, Aug 04 2021 *)

Equals log_2(1+sqrt(3)).

cp's OEIS Frontend

A346640 Decimal expansion of 2 - log_3(2).

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