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A289671

Consider the Post tag system defined in A284116; a(n) = number of binary words of length n which terminate in a cycle.

0, 2, 4, 8, 16, 48, 64, 128, 320, 704, 1536, 3328, 5632, 9728, 20480, 44032, 94208, 180224, 348160, 720896, 1458176, 2801664, 6062080, 12582912, 23986176, 49807360, 103809024, 202899456, 415760384, 853540864, 1663041536, 3332374528, 6752829440, 13153337344, 26055016448

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OFFSET

1,2

COMMENTS

For n such that no binary word of length n has an infinite orbit under the Post tag system (cf. A284116), which includes all n <= 57, a(n) + A289670(n) = 2^n.

LINKS

Don Reble, Table of n, a(n) for n = 1..57

EXAMPLE

For length n=2, there are two words which cycle, 10 and 11: 10 -> 101 -> 1101 -> 11101 -> 011101 -> 10100 -> 001101 -> 10100, which has entered a cycle.

MAPLE

See A289670.

MATHEMATICA

Table[ne = 0;

For[i = 0, i < 2^n, i++, lst = {};

w = IntegerString[i, 2, n];

While[! MemberQ[lst, w],

AppendTo[lst, w];

If[w == "", ne++; Break[]];

If[StringTake[w, 1] == "0", w = StringDrop[w <> "00", 3],

w = StringDrop[w <> "1101", 3]]]];

2^n - ne, {n, 1, 12}] (* Robert Price, Sep 26 2019 *)