cp's OEIS Frontend

A289671 Consider the Post tag system defined in A284116; a(n) = number of binary words of length n which terminate in a cycle.

%I A289671 #32 Aug 09 2025 19:41:36
%S A289671 0,2,4,8,16,48,64,128,320,704,1536,3328,5632,9728,20480,44032,94208,
%T A289671 180224,348160,720896,1458176,2801664,6062080,12582912,23986176,
%U A289671 49807360,103809024,202899456,415760384,853540864,1663041536,3332374528,6752829440,13153337344,26055016448
%N A289671 Consider the Post tag system defined in A284116; a(n) = number of binary words of length n which terminate in a cycle.
%C A289671 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.
%H A289671 Don Reble, <a href="/A289671/b289671.txt">Table of n, a(n) for n = 1..57</a>
%e A289671 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.
%p A289671 See A289670.
%t A289671 Table[ne = 0;
%t A289671 For[i = 0, i < 2^n, i++, lst = {};
%t A289671   w = IntegerString[i, 2, n];
%t A289671   While[! MemberQ[lst, w],
%t A289671    AppendTo[lst, w];
%t A289671    If[w == "", ne++; Break[]];
%t A289671    If[StringTake[w, 1] == "0", w = StringDrop[w <> "00", 3],
%t A289671     w = StringDrop[w <> "1101", 3]]]];
%t A289671 2^n - ne, {n, 1, 12}] (* _Robert Price_, Sep 26 2019 *)
%Y A289671 Cf. A284116, A284119, A284121, A289670-A289674.
%Y A289671 A289675 lists the initial words that terminate at the empty string.
%K A289671 nonn
%O A289671 1,2
%A A289671 _N. J. A. Sloane_, Jul 29 2017