A284116 -id:A284116 - cp's OEIS frontend

A291802 Take n-th word over {1,2} listed in A291797 and apply the Post tag system described in A284116 (but adapted to the alphabet {1,2}); a(n) = position of the longest word in the orbit, or -1 if the orbit is unbounded.

0, 3, 0, 0, 16, 14, 0, 0, 0, 13, 15, 11, 23, 9, 0, 18, 0, 22, 20, 0, 12, 8, 24, 10, 20, 18, 2, 6, 106, 16, 0, 23, 27, 0, 0, 0, 27, 5, 25, 0, 0, 105, 3, 103, 37, 15, 1, 1, 107, 17, 19, 101, 9, 99, 2, 35, 7, 7, 31, 13, 71, 97, 0, 0, 0, 0, 0, 22, 0, 34, 0, 0, 0

Offset: 1

N. J. A. Sloane, Sep 04 2017

nonn

Post's tag system maps a word w over {1,2} to w', where if w begins with 1, w' is obtained by appending 11 to w and deleting the first three letters, or if w begins with 2, w' is obtained by appending 2212 to w and deleting the first three letters.
We work over {1,2} rather than the official alphabet {0,1} because of the prohibition in the OEIS of terms (other than 0 itself) which begin with 0.
This is an analog of A291796 for the words in A291797.

Lars Blomberg, Table of n, a(n) for n = 1..8190 (For terms in A291797 up to length 36.)
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102. See Table 3.

Cf. A284116, A291794, A291795, A291796, A291797.

a(31)-a(73) from Lars Blomberg, Sep 08 2017

A284119 Preperiod (or threshold) of orbit of Post's {00, 1101} tag system applied to the word (100)^n, or -1 if this word has an unbounded trajectory.

Original entry on oeis.org

4, 15, 10, 25, 411, 47, 2128, 853, 372, 2805, 366, 2603, 703, 37912, 612, 127, 998, 2401, 1200, 623, 5280, 1778, 1462, 4346269, 4129, 3241, 7018, 3885, 14632, 7019, 4564, 4277, 147688, 1857, 11120, 81141, 20204, 3847, 116014, 7635, 6488, 5665, 6142, 73515, 5826, 6062, 3781, 7865, 28630

Offset: 1

Jeffrey Shallit, Mar 20 2017

nonn

Post's tag system 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 1101 to w and deleting the first three letters.
The empty word is included in the count.
The orbit of the word (100)^n for n=110 dies after 43913328040672 iterations. The longest word in the orbit is 31299218, which appeared at iteration 14392308412264. See also A291792. - Lars Blomberg, Oct 04 2017

			For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6:
Preperiod:
100100,
1001101,
11011101,
111011101,
0111011101,
101110100,
1101001101,
10011011101,
110111011101,
1110111011101,
01110111011101,
1011101110100,
11011101001101,
111010011011101,
0100110111011101,
followed by the period:
(011011101110100,
01110111010000,
1011101000000,
11010000001101,
100000011011101,
0000110111011101)
(repeat)

Lars Blomberg, Table of n, a(n) for n = 1..6075
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(110)
Lars Blomberg, Graph of preperiod of a(4974)
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A284116, A284121, A292090.

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)

Edited by N. J. A. Sloane, Jul 29 2017. Added "escape clause" to the definition, Apr 19 2018.

A284121 Period of orbit of Post's tag system applied to the word (100)^n (version 1), or -1 if the orbit increases without limit.

Original entry on oeis.org

2, 6, 6, 6, 1, 10, 28, 6, 10, 6, 6, 6, 1, 1, 6, 28, 10, 6, 10, 6, 6, 1, 6, 6, 1, 6, 6, 6, 6, 6, 6, 52, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 28, 6, 1, 1, 28, 6, 6, 6, 6, 6, 1, 6, 6, 6, 10, 6, 6, 6, 6, 1, 6, 1, 6, 6, 6, 6, 1, 6, 6, 6, 1, 6, 6, 6, 1, 10, 1, 10, 6, 6

Offset: 1

Jeffrey Shallit, Mar 20 2017

nonn

Post's tag system 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 1101 to w and deleting the first three letters.
The empty word is included in the count.
Here a(n)=1 if the orbit ends at the empty word. On the other hand, Asveld defines a(n) to be zero if that happens, which gives a different sequence, A291793. - N. J. A. Sloane, Sep 04 2017

			For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6. So a(2) = 6.

Lars Blomberg, Table of n, a(n) for n = 1..6075
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.

Cf. A284116, A284119, A291792, A291793.

Edited by N. J. A. Sloane, Jul 29 2017

a(50)-a(83) from Lars Blomberg, Sep 08 2017

A291067 Largest finite number of distinct words arising in Watanabe's tag system {00, 1011} applied to a binary word w, over all starting words w of length n.

Original entry on oeis.org

6, 5, 177, 178, 175, 174, 177, 178, 179, 180, 171, 550, 551, 548, 545, 550, 549, 610, 611, 608, 603, 14864, 14863, 14870, 14875, 14876, 15583, 15594, 15741, 15744, 15745, 15742, 15745, 15746, 15743, 114886, 114887, 114884, 114887, 114888, 114885, 404986

Offset: 1

N. J. A. Sloane, Aug 18 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.
Up through length 60, all starting strings either reach the empty word or enter a loop. - Don Reble, Sep 01 2017

			Examples of strings that achieve these records: "1", "10", "100", "0001", "10010", "100000", "1000000".
For example, at length 3, the trajectory of 100 begins 100, 1011, 11011, 111011, 0111011, 101100, 1001011, 10111011, 110111011, 1110111011, 01110111011, 1011101100, 11011001011, ..., and goes for 177 steps before a terms is repeated (at the 178-th step). So a(3) = 177. See A291075 for the full trajectory.

Lars Blomberg, Table of n, a(n) for n = 1..50
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]
N. J. A. Sloane, Maple programs that compute first 7 terms for each of A284116, A291067, A291068, A291069

For the 3-shift tag systems {00,1101}, {00, 1011}, {00, 1110}, {00, 0111} see A284116, A291067, A291068, A291069 respectively (as well as the cross-referenced entries mentioned there).

Cf. A291072, A291075, A291780, A291781.

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See link.
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a(8)-(42) from Lars Blomberg, Sep 16 2017

A291794 Consider Post's tag system 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

5, 15, 15, 19, 0, 31, 85, 37, 31, 37, 55, 37, 0, 0, 91, 85, 31, 127, 31, 33, 37, 0, 37, 37, 0, 73, 73, 163, 55, 19, 73, 157, 37, 73, 37, 37, 37, 163, 55, 37, 163, 37, 37, 85, 37, 0, 0, 85, 37, 127, 91, 37, 37, 0, 37, 69, 19, 31, 163, 163, 87, 55, 0, 37, 0, 55

Offset: 1

N. J. A. Sloane, Sep 04 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..6075
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.

Cf. A284116 and the cross-references there.

a(33) and beyond from Lars Blomberg, Apr 19 2018

A291792 Numbers m such that Post's tag system started at the word (100)^m eventually dies (i.e., reaches the empty string).

Original entry on oeis.org

5, 13, 14, 22, 25, 46, 47, 54, 63, 65, 70, 74, 78, 80, 91, 93, 106, 110, 117, 118, 128, 144, 148, 160, 166, 169, 190, 195, 199, 209, 222, 229, 234, 236, 239, 240, 243, 252, 254, 263, 264, 265, 266, 278, 281, 283, 286, 302, 304, 310, 324, 326, 327, 336, 339

Offset: 1

N. J. A. Sloane, Sep 04 2017

nonn

These are the numbers m such that A291793(m)=0, or equivalently A284121(m)=1.
Comments from Lars Blomberg on the method used in calculating the terms, Sep 14 2017: (Start)
Here is an overview of the method I have been using.
Build the words in a large byte array. Each iteration just adds 00 or 1101 to the end and removes 3 bytes from the beginning, without moving the whole word, just keeping track of the length of the word and where it starts within the array.
When the word reaches the end of the array it is moved to the beginning. This allows for very fast iterations, as long as the word is substantially shorter than the array.
The size of the byte array is 10^9, this is the longest word we can handle.
As for cycle detection, the words at iterations A: k*10^5 and B: (k+1)*10^5 are saved.
For iterations above B when the current word has the same length as B (a fast test), then check if the current word is equal to the one at B. If so, we have a cycle whose length can be determined simply by continued iterating. When the current iteration reaches C: (k+2)*10^5, move B->A, C->B, and continue.
The cycle has started somewhere between A and B and we know the cycle length. So restart two iterations at A and initially iterate one of them by the cycle length. Then iterate the two in parallel (being a cycle length apart) until they are equal, which gives the start of the cycle. Only the two words being iterated need to be stored.
One drawback with this method is that it cannot detect a cycle longer than 10^5 (or whatever value we choose). In that case the iterations will go on forever.
(End)
The trajectory of the word (100)^m for m=110 dies after 43913328040672 iterations, so 110 is a term in this sequence. The longest word in the trajectory is 31299218, which appeared at iteration 14392308412264. - Lars Blomberg, Oct 04 2017

Lars Blomberg, Table of n, a(n) for n = 1..1100
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
N. J. A. Sloane, Maple code for A291792, A284119, A291793, A284121, A291794, A291795, A291796, A292089, A292090, A292091, A292092, A292093, A292094.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part 1, Part 2, Slides. (Mentions this sequence)
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, A284121, A291793.

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)-a(17) from Lars Blomberg, Sep 08 2017

a(18)-a(55) from Lars Blomberg, Oct 15 2017

A291795 Consider Post's tag system 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

6, 16, 16, 22, 56, 34, 176, 76, 62, 208, 62, 208, 68, 768, 104, 88, 106, 224, 146, 134, 226, 172, 132, 4432, 206, 232, 378, 206, 432, 380, 208, 290, 1336, 224, 280, 1152, 336, 210, 1190, 356, 386, 292, 254, 806, 324, 362, 278, 316, 610, 1968, 710, 628, 10434

Offset: 1

N. J. A. Sloane, Sep 04 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..6075
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.

Cf. A284116 and the cross-references there.

a(33) and beyond from Lars Blomberg, Apr 19 2018

A291796 Consider Post's tag system 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

3, 14, 9, 16, 97, 34, 1293, 400, 91, 1734, 49, 1532, 51, 18168, 271, 78, 395, 674, 265, 260, 2701, 1068, 143, 935110, 2949, 1664, 2781, 2874, 9883, 3186, 1313, 996, 109875, 406, 5949, 57480, 15941, 258, 32359, 4712, 1223, 2424, 469, 35722, 1481, 1508, 395, 662

Offset: 1

N. J. A. Sloane, Sep 04 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..6075
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.

Cf. A284116 and the cross-references there.

a(33) and beyond from Lars Blomberg, Apr 19 2018

A291793 Period of orbit of Post's tag system applied to the word (100)^n (version 2), or -1 if the orbit increases without limit.

Original entry on oeis.org

2, 6, 6, 6, 0, 10, 28, 6, 10, 6, 6, 6, 0, 0, 6, 28, 10, 6, 10, 6, 6, 0, 6, 6, 0, 6, 6, 6, 6, 6, 6, 52, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 28, 6, 0, 0, 28, 6, 6, 6, 6, 6, 0, 6, 6, 6, 10, 6, 6, 6, 6, 0, 6, 0, 6, 6, 6, 6, 0, 6, 6, 6, 0, 6, 6, 6, 0, 10, 0, 10, 6, 6

Offset: 1

N. J. A. Sloane, Sep 04 2017, based on Jeffrey Shallit's A284121

nonn

Post's tag system 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 1101 to w and deleting the first three letters.
The empty word is included in the count.
Here, following Asveld, a(n)=0 if the orbit ends at the empty word. On the other hand, Shallit defines a(n) to be 1 if that happens, which gives a different sequence, A284121.
From A.H.M. Smeets, Jul 16 2020: (Start)
In general a tag as defined by Emil Leon Post, is given by a 4-tuple (Sigma,AF,n,w0), where Sigma is some (nonempty) alphabet, AF is the associated function (sometimes also called set of production rules) AF: Sigma -> Sigma*, n is the deletion number and w0 the initial string.
Here, the period lengths a(n) refer to the tags ({0,1},{(0,00),(1,1101)},3,100^n).
a(n) is an even number. Proof: for each cycle the number of associations (productions) 0 -> 00 must equal the number of associations (productions) 1 -> 1101 applied within a cycle. (End)

			For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6.

Lars Blomberg, Table of n, a(n) for n = 1..6075 (corrected for n=165 by _A.H.M. Smeets_)
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Lars Blomberg, Histogram over non-zero terms
Emil L. Post, Formal reductions of the general combinatorial decision problem., American Journal of Mathematics, Vol. 65, No. 2 (Apr., 1943), pp. 197-215.
Eric Weisstein's World of Mathematics, Tag System

Cf. A284116, A284119, A291792, A284121, A336287, A336327.

def step(w):
    i = 0
    while w[0] != alfabet[i]:
        i = i+1
    w = w+suffix[i]
    return w[n:len(w)]
alfabet, suffix, n, ws, w0, m = "01", ["00","1101"], 3, "100", "", 0
while m < 83:
    w0, m = w0+ws, m+1
    w, ww, i, a = w0, w0, 0, 0
    while w != "" and a == 0:
        w, i = step(w), i+1
        if i%1000 == 0:
            ww = w
        else:
            if w == ww or w == "":
                if w != "":
                    a = i%1000
                print(m,a) # A.H.M. Smeets, Jul 16 2020

a(50)-a(83) from Lars Blomberg, Sep 08 2017

A292091 Period of orbit of Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n.

Original entry on oeis.org

6, 6, 6, 6, 0, 518, 6, 518, 0, 6, 0, 6, 6, 28, 6, 0, 6, 34, 6, 0, 6, 0, 0, 6, 0, 518, 22, 22, 22, 6, 6, 6, 40, 518, 6, 6, 0, 0, 6, 6, 518, 518, 0, 518, 518, 6, 0, 6, 6, 26, 26, 6, 6, 6, 6, 6, 22, 6, 518, 6, 0, 16, 26, 0, 6, 0, 6, 0, 6, 6, 0, 6, 6, 6, 6, 6, 6

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)-(77) from Lars Blomberg, Sep 14 2017

cp's OEIS Frontend

A291802 Take n-th word over {1,2} listed in A291797 and apply the Post tag system described in A284116 (but adapted to the alphabet {1,2}); a(n) = position of the longest word in the orbit, or -1 if the orbit is unbounded.

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A284119 Preperiod (or threshold) of orbit of Post's {00, 1101} tag system applied to the word (100)^n, or -1 if this word has an unbounded trajectory.

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A284121 Period of orbit of Post's tag system applied to the word (100)^n (version 1), or -1 if the orbit increases without limit.

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A291067 Largest finite number of distinct words arising in Watanabe's tag system {00, 1011} applied to a binary word w, over all starting words w of length n.

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A291794 Consider Post's tag system 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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A291792 Numbers m such that Post's tag system started at the word (100)^m eventually dies (i.e., reaches the empty string).

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A291795 Consider Post's tag system 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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A291796 Consider Post's tag system 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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A291793 Period of orbit of Post's tag system applied to the word (100)^n (version 2), or -1 if the orbit increases without limit.

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A292091 Period of orbit of Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n.