A007061 The Ehrenfeucht-Mycielski sequence (1,2-version): a maximally unpredictable sequence.

1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn

Comments

Klaus Sutner remarks (Jun 26 2006) that this sequence is very similar to the Kimberling sequence A079101. Both sequences have every finite binary word as a factor; in fact, essentially the same proof works for both sequences.

Sutner continues: All words of length k seem to appear in the first 2^{k+2} bits. This is true for the first billion bits of the sequence, but no proof is known. The main open problem is whether the limiting density of 0's is 1/2. It seems to require a large amount of effort just to show that it is bounded away from 0, never mind some of the more exotic properties of the sequence (see the Sutner reference).

Start with a single bit 0. If the first n bits U(n) = a(1)a(2)...a(n) have already been chosen, let v be the longest suffix of U(n) that already appears in U(n-1). Find the last occurrence of v in U(n-1) and let b the bit that occurs immediately after. Then a(n+1) is the complement of b. (The entry gives the bits as 1's and 2s instead of 0's and 1's - compare A038219) - Joshua Zucker, Aug 11 2006

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Joshua Zucker, Table of n, a(n) for n = 1..1999
• A. Ehrenfeucht and J. Mycielski, A pseudorandom sequence - how random is it?, Amer. Math. Monthly, 99 (1992), 373-375.
• J. C. Kieffer and W. Szpankowski, On the Ehrenfeucht-Mycielski balance conjecture, Discrete Mathematics and Theoretical Computer Science, Proceedings, 2007 Conference on Analysis of Algorithms, AofA 07, (pp. 19-30).
• Terry McConnell, The Ehrenfeucht-Mycielski Sequence
• K. Sutner, The Ehrenfeucht-Mycielski sequence, 2001
• K. Sutner, The Ehrenfeucht-Mycielski sequence, 2001 [Cached copy]
• Wikipedia, Ehrenfeucht-Mycielski sequence

Crossrefs

Cf. A038219 (0-1 version), A079101.

Cf. A201881 (run lengths).

Cf. also A253059, A253060, A253061.

Programs

Haskell

a007061 n = a007061_list !! (n-1)
a007061_list = 1 : f [1] 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 = 3 - head ys
                         | otherwise          = b xyss
     vs = fromJust $ find (`isInfixOf` init us) $ tails us
-- Reinhard Zumkeller, Dec 05 2011

Python

from itertools import count, islice
def agen():
    astr, preval = "121", 2
    yield from [1, 2, 1]
    while True:
        an = 3 - 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 from 0 to 1, Aug 18 2006