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Search: id:A090822
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| A090822 |
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Gijswijt's sequence: a(1) = 1; for n>1, a(n) = largest integer k such that the word a(1)a(2)...a(n-1) is of the form xy^k for words x and y (where y has positive length), i.e. the maximal number of repeating blocks at the end of the sequence so far. |
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53
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| 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 1
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Here xy^k means the concatenation of the words x and k copies of y.
The name "Gijswijt's sequence" is due to njas, not the author!
Fix n and suppose a(n) = k. Let len_y(n) = length of shortest y for this k, and let len_x = n-1 - k*len_y(n) = corresponding length of x. A091407 and A091408 give len_y and len_x. For the subsequence when len_x = 0 see A091410 and A091411.
The first 4 occurs at a(220) (see A091409). The first time N appears is at about position 2^(2^(3^(4^(5^...^(N-1))))). - njas and Allan Wilks, Mar 14 2004
Mar 04 2004: Allan Wilks observes that in the first 100000 terms the fraction of [1's, 2's, 3's, 4's] seems to converge, to about [.287, .530, .179, .005] respectively.
When k=12 is reached, say, it is treated as the number 12, not as 1,2. This is not a base-dependent sequence.
Does this sequence have a finite average? Does anyone know the exact value? - Franklin T. Adams-Watters (franktaw(AT)netscape.net), Jan 24 2008. Reply from Maximilian Hasler (maximilian.hasler(AT)gmail.com): Given the cited observation "...the fraction of [1's, 2's, 3's, 4's] seems to converge, to about [.287, .530, .179, .005]..." that average should be the dot product of these vectors, i.e. about 1.904.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..24000
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
N. J. A. Sloane, Seven Staggering Sequences.
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CROSSREFS
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Cf. A091407, A091408, A091409 (records), A091410, A091411, A091579, A091586, A091970, A093955-A093958.
A091412 gives lengths of runs. A091413 gives partial sums.
Generalizations: A094781, A091975, A091976, A092331-A092335.
Sequence in context: A025829 A029285 A053633 this_sequence A091975 A091976 A094839
Adjacent sequences: A090819 A090820 A090821 this_sequence A090823 A090824 A090825
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KEYWORD
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nonn,nice
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AUTHOR
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Dion Gijswijt (gijswijt(AT)science.uva.nl), Feb 27 2004
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