A Slow-Growing Sequence Defined by an Unusual Recurrence
Fokko J. van de Bult and Dion C. Gijswijt
Korteweg-de Vries Institute for Mathematics
University of Amsterdam
Plantage Muidergracht 24
1018 TV Amsterdam
John P. Linderman, N. J. A. Sloane and Allan R. Wilks
AT&T Shannon Labs
180 Park Avenue
Florham Park, NJ 07932-0971
The sequence starts with
; to extend it one writes the sequence
so far as
are strings of integers,
is as large as possible: then the next term is
The sequence begins 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3,
appears for the first time at position 220, but a
does not appear until about position
The main result of the paper is a proof that the sequence is unbounded.
We also present results from extensive numerical investigations
of the sequence and of certain derived sequences, culminating
with a heuristic argument that
) appears for the first time at about position
The final section discusses generalizations.
Full version: pdf,
(Concerned with sequences
Received February 22 2006;
revised version received September 13 2006.
Published in Journal of Integer Sequences December 16 2006.
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