A289105 -id:A289105

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A289106

Positions of 1 in A289104; complement of A289105.

+20
3

3, 10, 14, 17, 18, 21, 24, 25, 28, 30, 35, 36, 39, 41, 46, 47, 49, 50, 53, 57, 59, 64, 65, 67, 68, 71, 75, 77, 81, 86, 87, 90, 93, 94, 96, 97, 100, 104, 106, 110, 115, 116, 119, 122, 123, 125, 126, 129, 132, 133, 136, 140, 142, 147, 148, 151, 153, 157, 162

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Conjecture: a(n)/n -> (3 + sqrt(5))/2 = 1 + golden ratio = 1 + tau, and 1 + tau < a(n)/n < 4 for n >= 4.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

MATHEMATICA

z = 11; (* number of iterates *)

s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];

w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "011", "10" -> "000"}]

TableForm[Table[w[n], {n, 0, 10}]]

st = ToCharacterCode[w[z]] - 48 (* A289104 *)

Flatten[Position[st, 0]] (* A289105 *)

Flatten[Position[st, 1]] (* A289106 *)

CROSSREFS

Cf. A001622, A289104, A289105, A289106.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 28 2017

STATUS

approved

A289104

Fixed point of the mapping 00->0010, 01->011, 10->000, starting with 00.

+10
4

0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Conjecture: the number of letters (0's and 1's) in the n-th iterate of the mapping is given by A289107. For connections to the golden ratio, see A289105 and A289106.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

EXAMPLE

The first six iterates of the mapping:

0010

0010000

001000000100

0010000001000100110010

0010000001000100110010011001010000110