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A038219 The Ehrenfeucht-Mycielski sequence (0,1-version): a maximally unpredictable sequence. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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 R. McConnell, DeBruijn Strings, double helices, and the Ehrenfeucht-Mycielski mechanism, arXiv:1303.6820 [math.CO], 2013.
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.
EXAMPLE
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
MAPLE
See Lunnon link.
PROG
(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
(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
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).
Sequence in context: A288633 A284775 A156259 * A330731 A138710 A356556
KEYWORD
nonn,nice
AUTHOR
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
STATUS
approved

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Last modified March 19 07:36 EDT 2024. Contains 370956 sequences. (Running on oeis4.)