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

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

1,2

Comments

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.

+---+---+---+---+

Links

Table of n, a(n) for n=1..105.

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.

Formula

Equals log_2(1+sqrt(3)).

Example

1.4499843134764958489211625600623791...

Mathematica

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

Crossrefs

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

Sequence in context: A175051 A011364 A016713 * A168038 A093995 A168280

Adjacent sequences: A343617 A343618 A343619 * A343621 A343622 A343623

Keyword

cons,nonn

Author

Kevin Ryde, Aug 04 2021

Status

approved