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

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

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

Comments

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

Examples

			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
		

Links

• 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

Crossrefs

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).

Programs

Haskell

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

See Links section.

Python

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

Extensions

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