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A135473 a(n) = number of strings of length n that can be obtained by starting with abc and repeatedly doubling any substring in place. 15
0, 0, 1, 3, 8, 21, 54, 138, 355, 924, 2432, 6461, 17301, 46657, 126656, 345972, 950611, 2626253, 7292268, 20342805, 56993909, 160317859, 452642235, 1282466920, 3645564511, 10395117584, 29727982740, 85251828792, 245120276345, 706529708510, 2041260301955, 5910531770835, 17149854645474, 49859456251401, 145223624492108, 423722038708874, 1238318400527185 (list; graph; refs; listen; history; text; internal format)



The problem can be restated as follows: look at the language L over {1,2,3}* which contains 123 and is closed under duplication. What is the growth function of L (or its subword complexity function)?

It is known that the language L is not regular [Wang]

Several generalizations suggest themselves: What if we start with k different letters (here k=3)? What if we start with k different letters and fix the number of duplications d? See A137739, A137740, A137741, A137742, A137743, A137744, A137745, A137746, A137747, A137748.


D. P. Bovet and S. Varricchio, On the regularity of languages on a binary alphabet generated by copying systems, Information Processing Letters, 44 (1992), 119-123.

Juergen Dassow, Victor Mitrana and Gheorghe Paun: On the Regularity of Duplication Closure. Bulletin of the EATCS 69 (1999), 133-136.

Ming-wei Wang, On the Irregularity of the Duplication Closure, Bulletin of the EATCS, Vol. 70, 2000, 162-163.


Table of n, a(n) for n=1..37.

Index entries for doubling substrings


Empirically, grows like 3^n.


n=3: abc

n=4: aabc, abbc, abcc

n=5: aaabc, abbbc, abccc, aabbc, aabcc, abbcc, ababc, abcbc


Cf. A135017, A135156, A135157, A135475, A135479, A130838.

Cf. also A137739, A137740, A137741, A137742, A137743, A137744, A137745, A137746, A137747, A137748.

Sequence in context: A094723 A127358 A077849 * A242452 A190139 A005580

Adjacent sequences:  A135470 A135471 A135472 * A135474 A135475 A135476




Max Alekseyev, Jan 07 2008


a(19) - a(33) from David Applegate, Feb 12 2008

Extended to 37 terms by David Applegate, Feb 16 2008

Thanks to Robert Mercas and others for comments and references - N. J. A. Sloane, Apr 26 2013



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Last modified July 8 03:38 EDT 2020. Contains 335504 sequences. (Running on oeis4.)