A308175 -id:A308175 - cp's OEIS frontend

A308174 Let EM denote the Ehrenfeucht-Mycielski sequence A038219, and let P(n) = [EM(1),...,EM(n)]. To compute EM(n+1) for n>=3, we find the longest suffix S (say) of P(n) which has previously appeared in P(n). Suppose the most recent appearance of S began at index n-t(n). Then a(n) = length of S, while t(n) is given in A308175.

1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 3

N. J. A. Sloane, May 21 2019, corrected and extended May 21 2019

nonn

Then EM(n+1) is the complement of the bit following the most recent appearance of S.

			Tableau showing calculation of terms 3 through 13
1   2   3   4   5   6   7   8   9  10  11  12  13  n
0   1   0   0   1   1   0   1   0   1   1   1   0  A038219(n)
-   -   0   0  01   1  10  01 010 101 011  11 110  S
-   -   1   1   2   1   2   2   3   3   3   2   3  s = A308174(n)
-   -   1   3   1   5   2   4   1   6   4  10   5  previous
-   -   2   1   4   1   5   4   8   4   7   2   8  t = A308175(n)
"Previous" = index of start of most recent previous occurrence of S; s = |S|; t = n - "previous" = A308175(n)

Rémy Sigrist, Table of n, a(n) for n = 3..50000
Rémy Sigrist, Perl program for A308174

Cf. A038219, A308175.

Perl
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See Links section.
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More terms from Rémy Sigrist, May 21 2019

A038219 The Ehrenfeucht-Mycielski sequence (0,1-version): a maximally unpredictable sequence.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mira Bernstein

The sequence starts 0,1,0 and continues according to the following rule: find the longest suffix that has occurred at least once previously. If there is more than one previous occurrences select the most recent one. The next digit of the sequence is the opposite of the one following the previous occurrence. - Christopher Carl Heckman, Feb 10 2005

			We start with a(1)=0, a(2)=1, a(3)=0.
The longest suffix we have seen before is "0", when it occurred at the start, followed by 1. So a(4) = 0. We now have 0100.
The longest suffix we have seen before is again "0", when it occurred at a(3), followed by a(4)=0. So a(5) = 1. We now have 01001.
The longest suffix we have seen before is "01", when it occurred at a(1),a(2), followed by a(3)=0. So a(6) = 1. We now have 010011.
And so on.
For further illustrations of calculating these terms, see A308174 and A308175. - _N. J. A. Sloane_, May 21 2019

Rémy Sigrist, Table of n, a(n) for n = 1..100000 (first 5000 terms from Reinhard Zumkeller)
A. Ehrenfeucht and J. Mycielski, A pseudorandom sequence - how random is it?, Amer. Math. Monthly, 99 (1992), 373-375.
Grzegorz Herman and Michael Soltys, On the Ehrenfeucht-Mycielski sequence, Journal of Discrete Algorithms, 7, No. 4 (2009), 500-508.
J. C. Kieffer and W. Szpankowski, On the Ehrenfeucht-Mycielski balance conjecture. Discrete Mathematics and Theoretical Computer Science (2007), 19-30.
Fred Lunnon, Maple Program for A038219 (Complexity about O(n log n) arithmetic operations)
Terry McConnell, The Ehrenfeucht-Mycielski Sequence
Terry R. McConnell, DeBruijn Strings, double helices, and the Ehrenfeucht-Mycielski mechanism, arXiv:1303.6820 [math.CO], 2013.
Rémy Sigrist, Perl program for A038219
K. Sutner, The Ehrenfeucht-Mycielski sequence, 2001 [broken link]
K. Sutner, The Ehrenfeucht-Mycielski sequence, 2001 [Cached copy]
Klaus Sutner, The Ehrenfeucht-Mycielski Sequence, LNCS 2759 (2003) 282-293.
Wikipedia, Ehrenfeucht-Mycielski sequence

Cf. A007061 (1, 2 version).
Cf. A201881 (run lengths).
Cf. also A253059, A253060, A253061.
For first appearance of subwords see A308173.
A308174, A308175 are used in the calculation of a(n).

a038219 n = a038219_list !! n
a038219_list = 0 : f [0] where
   f us = a' : f (us ++ [a']) where
        a' = b $ reverse $ map (`splitAt` us) [0..length us - 1] where
           b ((xs,ys):xyss) | vs `isSuffixOf` xs = 1 - head ys
                            | otherwise          = b xyss
        vs = fromJust $ find (`isInfixOf` init us) $ tails us
-- Reinhard Zumkeller, Dec 05 2011

Maple
```
See Lunnon link.
```
Perl
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See Links section.
```

from itertools import count, islice
def agen():
    astr, preval = "010", 1
    yield from [0, 1, 0]
    while True:
        an = 1 - preval
        yield an
        astr += str(an)
        for l in range(len(astr)-1, 0, -1):
            idx = astr.rfind(astr[-l:], 0, len(astr)-1)
            if idx >= 0: preval = int(astr[idx+l]); break
print(list(islice(agen(), 105))) # Michael S. Branicky, Aug 03 2022

More terms from Joshua Zucker, Aug 11 2006
Offset changed by Reinhard Zumkeller, Dec 11 2011
Edited by N. J. A. Sloane, May 12 2019

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