A284121 -id:A284121 - cp's OEIS frontend

A289677 a(n) = A289671(n)/2^f(n), where f(n) = 2*floor((n-1)/3) + ((n+2) mod 3) = A004523(n).

0, 1, 1, 2, 2, 3, 4, 4, 5, 11, 12, 13, 22, 19, 20, 43, 46, 44, 85, 88, 89, 171, 185, 192, 366, 380, 396, 774, 793, 814, 1586, 1589, 1610, 3136, 3106, 3130, 6123, 6078, 6103, 12088, 12147, 12229, 24283, 24534, 24736, 49040, 49482, 49879, 99031, 99792, 100747, 200444, 201892, 203765, 405931, 408478, 411403

Offset: 1

N. J. A. Sloane, Aug 01 2017

nonn

This is the number of distinct binary words w of length n that eventually cycle under the Post tag system (see A284116, A289670) reduced to take into account the observation made by Don Reble that (if the bits of w are labeled from the left starting at bit 0) bits 1,2,4,5,7,8,... (not a multiple of 3) are "junk DNA" and have no effect on the outcome.

Cf. A284116, A284119, A284121, A289670, A289671, A289672, A289673, A289674, A289675, A289676.

Cf. also A290436-A290441.

from _future_ import division
def A289677(n):
    c, k, r, n2, cs, ts = 0, 1+(n-1)//3, 2**((n-1) % 3), 2**(n-1), set(), set()
    for i in range(2**k):
        j, l = int(bin(i)[2:],8)*r, n2
        traj = set([(l,j)])
        while True:
            if j >= l:
                j = j*16+13
                l *= 2
            else:
                j *= 4
                l //= 2
            if l == 0:
                ts |= traj
                break
            j %= 2*l
            if (l,j) in traj:
                c += 1
                cs |= traj
                break
            if (l,j) in cs:
                c += 1
                break
            if (l,j) in ts:
                break
            traj.add((l,j))
    return c # Chai Wah Wu, Aug 03 2017

Corrected by Don Reble, Aug 01 2017 (there were errors in A289671).

A290436 a(n) = A289676(3*n+1).

Original entry on oeis.org

2, 2, 4, 5, 10, 21, 43, 85, 146, 250, 462, 960, 2069, 4296, 8485, 16496, 32041, 61700, 118357

Offset: 0

N. J. A. Sloane, Aug 02 2017

No formulas or recurrences are known for the important sequences A289670 and A289671. The essence of these two sequences is captured in the six entries A290436-A290441. Any numerical properties of these would be most welcome.

Cf. A284116, A284119, A284121, A289670-A289677, A290436-A290441.

A290441 a(n) = A289677(3*n).

Original entry on oeis.org

1, 3, 5, 13, 20, 44, 89, 192, 396, 814, 1610, 3130, 6103, 12229, 24736, 49879, 100747, 203765, 411403

Offset: 1

N. J. A. Sloane, Aug 02 2017

No formulas or recurrences are known for the important sequences A289670 and A289671. The essence of these two sequences is captured in the six entries A290436-A290441. Any numerical properties of these would be most welcome.

Cf. A284116, A284119, A284121, A289670-A289677, A290436-A290441.

A292090 Preperiod (or threshold) of orbit of Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n.

Original entry on oeis.org

171, 166, 11, 154, 105, 14, 57, 68, 173, 1098, 8265, 720, 1715, 130, 1979, 2024, 833, 162, 591, 6124, 59673, 748, 11631, 3200, 1453, 13740, 2947, 2202, 15101, 1268, 608049, 30758, 29903, 1076, 17547, 2888, 72231, 10154, 2321, 68916, 10965, 2276, 151785, 4678

Offset: 1

N. J. A. Sloane, Sep 10 2017

nonn

Watanabe's tag system {00/1011} maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1011 to w and deleting the first three letters.

The empty word is included in the count.

			The following is the analog of columns 3 through 7 of Asveld's Table 1.
1 [171, 6, 56, 59, 138]
2 [166, 6, 56, 59, 133]
3 [11, 6, 16, 17, 10]
4 [154, 6, 56, 59, 121]
5 [105, 0, 0, 31, 24]
6 [14, 518, 28, 85, 215]
7 [57, 6, 38, 41, 36]
8 [68, 518, 42, 85, 333]
9 [173, 0, 0, 49, 38]
10 [1098, 6, 34, 159, 407]
11 [8265, 0, 0, 328, 4429]
12 [720, 6, 34, 93, 343]
13 [1715, 6, 34, 93, 1338]
14 [130, 28, 82, 83, 85]
15 [1979, 6, 20, 215, 720]
16 [2024, 0, 0, 193, 1023]
17 [833, 6, 70, 121, 420]
18 [162, 34, 100, 101, 105]
19 [591, 6, 20, 109, 118]
20 [6124, 0, 0, 357, 2259]
21 [59673, 6, 20, 781, 33530]
22 [748, 0, 0, 150, 328]
23 [11631, 0, 0, 273, 6250]
24 [3200, 6, 56, 261, 1515]
...

Lars Blomberg, Table of n, a(n) for n = 1..6080
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Lars Blomberg, Graph of the terms
Lars Blomberg, Graph of preperiod of a(2263)
Lars Blomberg, Graph of preperiod of a(2540)
Shigeru Watanabe, Periodicity of Post's normal process of tag, in Jerome Fox, ed., Proceedings of Symposium on Mathematical Theory of Automata, New York, April 1962, Polytechnic Press, Polytechnic Institute of Brooklyn, 1963, pp. 83-99. [Annotated scanned copy]

Cf. A284116, A284119, A291067, A291780, A291781.

Asveld's Table 1 gives data about the behavior of Post's 3-shift tag system {00/1101} applied to the word (100)^n. The first column gives n, the nonzero values in column 2 give A291792, and columns 3 through 7 give A284119, 291793 (or A284121), A291794, A291795, A291796. For the corresponding data for Watanabe's 3-shift tag system {00/1011} applied to (100)^n see A292089, A292090, A292091, A292092, A292093, A292094.

From Lars Blomberg, Apr 20 2018: (Start)
Using Excel, trendlines were created for the preperiod of the Post Tag and Watanabe Tag systems as follows:
A284119: y = 8.6528*x^2.0831, R^2 = 0.478.
A292090: y = 8.5595*x^2.1033, R^2 = 0.472.
Although the error value is rather large, the curves are quite similar. (End)

a(25)-(44) from Lars Blomberg, Sep 14 2017

A289672 Consider the Post tag system defined in A284116; a(n) = maximum, taken over all binary words w of length n which terminate in a cycle, of the number of words in the orbit of w.

Original entry on oeis.org

4, 3, 4, 7, 8, 7, 14, 15, 14, 15, 16, 15, 24, 25, 28, 29, 30, 35, 38, 39, 38, 39, 38

Offset: 1

N. J. A. Sloane, Jul 29 2017

The terminating empty word is included in the count.

			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.

Cf. A284116, A284119, A284121, A289670, A289671, A289673, A289674, A289675.

# Uses procedures f1 and P from A289670.
# Count strings of length n which terminate and which cycle
# Print max length to reach empty word (mx)
mx:=[];
for n from 1 to 11 do
lprint("starting length ",n);
m:=0;
for n1 from 0 to 2^n-1 do
t1:=convert(2^n+n1,base,2); t2:=[seq(t1[i],i=1..n)];
map(x->convert(x,string),t2); t3:=Join(%,""); t4:=P(%);
if t4 <> 999 then if t4>m then m:=t4; fi; fi;
od;
mx:=[op(mx),m];
od:
mx;

a(12)-a(23) from Indranil Ghosh, Jul 30 2017

A292089 Numbers n such that Watanabe's 3-shift tag system {00/1011} started at the word (100)^n eventually dies (i.e., reaches the empty string).

Original entry on oeis.org

5, 9, 11, 16, 20, 22, 23, 25, 37, 38, 43, 47, 61, 64, 66, 68, 71, 82, 87, 95, 100, 115, 119, 120, 123, 126, 137, 141, 142, 143, 144, 147, 149, 153, 156, 158, 164, 165, 171, 178, 179, 183, 188, 195, 196, 201, 202, 203, 205, 206, 212, 214, 216, 218, 223, 232

Offset: 1

N. J. A. Sloane, Sep 10 2017

nonn

Watanabe's tag system {00/1011} maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1011 to w and deleting the first three letters.
These are the numbers such that A292091(n)=0.
Oct 11, 2017: Lars Blomberg has found that 872 is a member of this sequence. The word (100)^872 reaches the empty string after 72392976118788 iterations. The attached graph shows the lengths of the successive words in the trajectory. - N. J. A. Sloane, Oct 13 2017

			The following is the analog of columns 3 through 7 of Asveld's Table 1.
1 [171, 6, 56, 59, 138]
2 [166, 6, 56, 59, 133]
3 [11, 6, 16, 17, 10]
4 [154, 6, 56, 59, 121]
5 [105, 0, 0, 31, 24]
6 [14, 518, 28, 85, 215]
7 [57, 6, 38, 41, 36]
8 [68, 518, 42, 85, 333]
9 [173, 0, 0, 49, 38]
10 [1098, 6, 34, 159, 407]
11 [8265, 0, 0, 328, 4429]
12 [720, 6, 34, 93, 343]
13 [1715, 6, 34, 93, 1338]
14 [130, 28, 82, 83, 85]
15 [1979, 6, 20, 215, 720]
16 [2024, 0, 0, 193, 1023]
17 [833, 6, 70, 121, 420]
18 [162, 34, 100, 101, 105]
19 [591, 6, 20, 109, 118]
20 [6124, 0, 0, 357, 2259]
21 [59673, 6, 20, 781, 33530]
22 [748, 0, 0, 150, 328]
23 [11631, 0, 0, 273, 6250]
24 [3200, 6, 56, 261, 1515]
...

Lars Blomberg, Table of n, a(n) for n = 1..1440
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Lars Blomberg, Lengths of words in the trajectory of (100)^872
N. J. A. Sloane, Maple code for A291792, A284119, A291793, A284121, A291794, A291795, A291796, A292089, A292090, A292091, A292092, A292093, A292094.
Shigeru Watanabe, Periodicity of Post's normal process of tag, in Jerome Fox, ed., Proceedings of Symposium on Mathematical Theory of Automata, New York, April 1962, Polytechnic Press, Polytechnic Institute of Brooklyn, 1963, pp. 83-99. [Annotated scanned copy]

Cf. A284116, A291067, A291780, A291781.

Asveld's Table 1 gives data about the behavior of Post's 3-shift tag system {00/1101} applied to the word (100)^n. The first column gives n, the nonzero values in column 2 give A291792, and columns 3 through 7 give A284119, A291793 (or A284121), A291794, A291795, A291796. For the corresponding data for Watanabe's 3-shift tag system {00/1011} applied to (100)^n see A292089, A292090, A292091, A292092, A292093, A292094.

a(8)-(18) from Lars Blomberg, Sep 14 2017

a(19) and beyond from Lars Blomberg, Apr 20 2018

A292092 Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = length of first word we see that is in the cycle, if the orbit cycles, or 0 if the orbit reaches the empty string, or -1 if the orbit is unbounded.

Original entry on oeis.org

56, 56, 16, 56, 0, 28, 38, 42, 0, 34, 0, 34, 34, 82, 20, 0, 70, 100, 20, 0, 20, 0, 0, 56, 0, 46, 64, 64, 64, 92, 74, 34, 118, 66, 88, 52, 0, 0, 34, 268, 42, 34, 0, 46, 30, 92, 0, 16, 34, 76, 76, 34, 34, 38, 110, 20, 64, 92, 46, 56, 0, 46, 76, 0, 74, 0, 88, 0

Offset: 1

N. J. A. Sloane, Sep 10 2017

nonn

Watanabe's tag system {00/1011} maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1011 to w and deleting the first three letters.
The empty word is included in the count.
Following Asveld we set a(n)=0 if the orbit ends at the empty word.

			The following is the analog of columns 3 through 7 of Asveld's Table 1.
1 [171, 6, 56, 59, 138]
2 [166, 6, 56, 59, 133]
3 [11, 6, 16, 17, 10]
4 [154, 6, 56, 59, 121]
5 [105, 0, 0, 31, 24]
6 [14, 518, 28, 85, 215]
7 [57, 6, 38, 41, 36]
8 [68, 518, 42, 85, 333]
9 [173, 0, 0, 49, 38]
10 [1098, 6, 34, 159, 407]
11 [8265, 0, 0, 328, 4429]
12 [720, 6, 34, 93, 343]
13 [1715, 6, 34, 93, 1338]
14 [130, 28, 82, 83, 85]
15 [1979, 6, 20, 215, 720]
16 [2024, 0, 0, 193, 1023]
17 [833, 6, 70, 121, 420]
18 [162, 34, 100, 101, 105]
19 [591, 6, 20, 109, 118]
20 [6124, 0, 0, 357, 2259]
21 [59673, 6, 20, 781, 33530]
22 [748, 0, 0, 150, 328]
23 [11631, 0, 0, 273, 6250]
24 [3200, 6, 56, 261, 1515]
...

Lars Blomberg, Table of n, a(n) for n = 1..6080
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Shigeru Watanabe, Periodicity of Post's normal process of tag, in Jerome Fox, ed., Proceedings of Symposium on Mathematical Theory of Automata, New York, April 1962, Polytechnic Press, Polytechnic Institute of Brooklyn, 1963, pp. 83-99. [Annotated scanned copy]

Cf. A284116, A291067, A291780, A291781.

Asveld's Table 1 gives data about the behavior of Post's 3-shift tag system {00/1101} applied to the word (100)^n. The first column gives n, the nonzero values in column 2 give A291792, and columns 3 through 7 give A284119, 291793 (or A284121), A291794, A291795, A291796. For the corresponding data for Watanabe's 3-shift tag system {00/1011} applied to (100)^n see A292089, A292090, A292091, A292092, A292093, A292094.

a(25)-(68) from Lars Blomberg, Sep 14 2017

A292093 Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = length of the longest word in the orbit, or -1 if the orbit is unbounded.

Original entry on oeis.org

59, 59, 17, 59, 31, 85, 41, 85, 49, 159, 328, 93, 93, 83, 215, 193, 121, 101, 109, 357, 781, 150, 273, 261, 171, 341, 182, 229, 551, 187, 2627, 593, 503, 187, 400, 261, 1369, 371, 226, 1045, 374, 280, 849, 375, 437, 255, 667, 365, 291, 2972, 463, 905, 631, 405

Offset: 1

N. J. A. Sloane, Sep 10 2017

nonn

Watanabe's tag system {00/1011} maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1011 to w and deleting the first three letters.

The empty word is included in the count.

			The following is the analog of columns 3 through 7 of Asveld's Table 1.
1 [171, 6, 56, 59, 138]
2 [166, 6, 56, 59, 133]
3 [11, 6, 16, 17, 10]
4 [154, 6, 56, 59, 121]
5 [105, 0, 0, 31, 24]
6 [14, 518, 28, 85, 215]
7 [57, 6, 38, 41, 36]
8 [68, 518, 42, 85, 333]
9 [173, 0, 0, 49, 38]
10 [1098, 6, 34, 159, 407]
11 [8265, 0, 0, 328, 4429]
12 [720, 6, 34, 93, 343]
13 [1715, 6, 34, 93, 1338]
14 [130, 28, 82, 83, 85]
15 [1979, 6, 20, 215, 720]
16 [2024, 0, 0, 193, 1023]
17 [833, 6, 70, 121, 420]
18 [162, 34, 100, 101, 105]
19 [591, 6, 20, 109, 118]
20 [6124, 0, 0, 357, 2259]
21 [59673, 6, 20, 781, 33530]
22 [748, 0, 0, 150, 328]
23 [11631, 0, 0, 273, 6250]
24 [3200, 6, 56, 261, 1515]
...

Lars Blomberg, Table of n, a(n) for n = 1..6080
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Shigeru Watanabe, Periodicity of Post's normal process of tag, in Jerome Fox, ed., Proceedings of Symposium on Mathematical Theory of Automata, New York, April 1962, Polytechnic Press, Polytechnic Institute of Brooklyn, 1963, pp. 83-99. [Annotated scanned copy]

Cf. A284116, A291067, A291780, A291781.

Asveld's Table 1 gives data about the behavior of Post's 3-shift tag system {00/1101} applied to the word (100)^n. The first column gives n, the nonzero values in column 2 give A291792, and columns 3 through 7 give A284119, 291793 (or A284121), A291794, A291795, A291796. For the corresponding data for Watanabe's 3-shift tag system {00/1011} applied to (100)^n see A292089, A292090, A292091, A292092, A292093, A292094.

a(25)-(54) from Lars Blomberg, Sep 14 2017

A292094 Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = position of the longest word in the orbit, or -1 if the orbit is unbounded.

Original entry on oeis.org

138, 133, 10, 121, 24, 215, 36, 333, 38, 407, 4429, 343, 1338, 85, 720, 1023, 420, 105, 118, 2259, 33530, 328, 6250, 1515, 370, 9729, 2059, 825, 6282, 309, 310620, 20089, 10014, 187, 12069, 1101, 21756, 2359, 1253, 53811, 7277, 598, 103772, 1275, 5584, 269

Offset: 1

N. J. A. Sloane, Sep 10 2017

nonn

Watanabe's tag system {00/1011} maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1011 to w and deleting the first three letters.

The empty word is included in the count.

			The following is the analog of columns 3 through 7 of Asveld's Table 1.
1 [171, 6, 56, 59, 138]
2 [166, 6, 56, 59, 133]
3 [11, 6, 16, 17, 10]
4 [154, 6, 56, 59, 121]
5 [105, 0, 0, 31, 24]
6 [14, 518, 28, 85, 215]
7 [57, 6, 38, 41, 36]
8 [68, 518, 42, 85, 333]
9 [173, 0, 0, 49, 38]
10 [1098, 6, 34, 159, 407]
11 [8265, 0, 0, 328, 4429]
12 [720, 6, 34, 93, 343]
13 [1715, 6, 34, 93, 1338]
14 [130, 28, 82, 83, 85]
15 [1979, 6, 20, 215, 720]
16 [2024, 0, 0, 193, 1023]
17 [833, 6, 70, 121, 420]
18 [162, 34, 100, 101, 105]
19 [591, 6, 20, 109, 118]
20 [6124, 0, 0, 357, 2259]
21 [59673, 6, 20, 781, 33530]
22 [748, 0, 0, 150, 328]
23 [11631, 0, 0, 273, 6250]
24 [3200, 6, 56, 261, 1515]
...

Lars Blomberg, Table of n, a(n) for n = 1..6080
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Shigeru Watanabe, Periodicity of Post's normal process of tag, in Jerome Fox, ed., Proceedings of Symposium on Mathematical Theory of Automata, New York, April 1962, Polytechnic Press, Polytechnic Institute of Brooklyn, 1963, pp. 83-99. [Annotated scanned copy]

Cf. A284116, A291067, A291780, A291781.

Asveld's Table 1 gives data about the behavior of Post's 3-shift tag system {00/1101} applied to the word (100)^n. The first column gives n, the nonzero values in column 2 give A291792, and columns 3 through 7 give A284119, 291793 (or A284121), A291794, A291795, A291796. For the corresponding data for Watanabe's 3-shift tag system {00/1011} applied to (100)^n see A292089, A292090, A292091, A292092, A292093, A292094.

a(25)-(46) from Lars Blomberg, Sep 14 2017

A289676 a(n) = A289670(n)/2^f(n), where f(n) = 2*floor((n-1)/3) + ((n+2) mod 3).

Original entry on oeis.org

2, 1, 1, 2, 2, 1, 4, 4, 3, 5, 4, 3, 10, 13, 12, 21, 18, 20, 43, 40, 39, 85, 71, 64, 146, 132, 116, 250, 231, 210, 462, 459, 438, 960, 990, 966, 2069, 2114, 2089, 4296, 4237, 4155, 8485, 8234, 8032, 16496, 16054, 15657, 32041, 31280, 30325, 61700, 60252, 58379, 118357, 115810, 112885

Offset: 1

N. J. A. Sloane, Aug 01 2017

nonn

This is the number of distinct binary words w of length n that terminate under the Post tag system (see A284116, A289670) reduced to take into account the observation made by Don Reble that (if the bits of w are labeled from the left starting at bit 0) bits 1,2,4,5,7,8,... (not a multiple of 3) are "junk DNA" and have no effect on the outcome.

Cf. A284116, A284119, A284121, A289670, A289671, A289672, A289673, A289674, A289675, A289677.

Cf. also A290436-A290441.

from _future_ import division
def A289676(n):
    c, k, r, n2, cs, ts = 0, 1+(n-1)//3, 2**((n-1) % 3), 2**(n-1), set(), set()
    for i in range(2**k):
        j, l = int(bin(i)[2:],8)*r, n2
        traj = set([(l,j)])
        while True:
            if j >= l:
                j = j*16+13
                l *= 2
            else:
                j *= 4
                l //= 2
            if l == 0:
                c += 1
                ts |= traj
                break
            j %= 2*l
            if (l,j) in traj:
                cs |= traj
                break
            if (l,j) in cs:
                break
            if (l,j) in ts:
                c += 1
                break
            traj.add((l,j))
    return c # Chai Wah Wu, Aug 03 2017

Corrected by Don Reble, Aug 01 2017 (there were errors in A289670)

cp's OEIS Frontend

A289677 a(n) = A289671(n)/2^f(n), where f(n) = 2*floor((n-1)/3) + ((n+2) mod 3) = A004523(n).

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A290436 a(n) = A289676(3*n+1).

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A290441 a(n) = A289677(3*n).

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A292090 Preperiod (or threshold) of orbit of Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n.

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A289672 Consider the Post tag system defined in A284116; a(n) = maximum, taken over all binary words w of length n which terminate in a cycle, of the number of words in the orbit of w.

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A292089 Numbers n such that Watanabe's 3-shift tag system {00/1011} started at the word (100)^n eventually dies (i.e., reaches the empty string).

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A292092 Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = length of first word we see that is in the cycle, if the orbit cycles, or 0 if the orbit reaches the empty string, or -1 if the orbit is unbounded.

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A292093 Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = length of the longest word in the orbit, or -1 if the orbit is unbounded.

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A292094 Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = position of the longest word in the orbit, or -1 if the orbit is unbounded.

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A289676 a(n) = A289670(n)/2^f(n), where f(n) = 2*floor((n-1)/3) + ((n+2) mod 3).

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