This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135473 #25 Jun 07 2025 15:57:00
%S A135473 0,0,1,3,8,21,54,138,355,924,2432,6461,17301,46657,126656,345972,
%T A135473 950611,2626253,7292268,20342805,56993909,160317859,452642235,
%U A135473 1282466920,3645564511,10395117584,29727982740,85251828792,245120276345,706529708510,2041260301955,5910531770835,17149854645474,49859456251401,145223624492108,423722038708874,1238318400527185
%N A135473 a(n) = number of strings of length n that can be obtained by starting with abc and repeatedly doubling any substring in place.
%C A135473 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)?
%C A135473 It is known that the language L is not regular [Wang]
%C A135473 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 A135473 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.
%D A135473 Juergen Dassow, Victor Mitrana and Gheorghe Paun: On the Regularity of Duplication Closure. Bulletin of the EATCS 69 (1999), 133-136.
%D A135473 Ming-wei Wang, On the Irregularity of the Duplication Closure, Bulletin of the EATCS, Vol. 70, 2000, 162-163.
%H A135473 Martin Fuller, <a href="/A135473/b135473.txt">Table of n, a(n) for n = 1..44</a>
%H A135473 <a href="/index/Do#repeat">Index entries for doubling substrings</a>
%F A135473 Binomial transform of A135017. - _Martin Fuller_, Jun 06 2025
%F A135473 Empirically, grows like 3^n.
%e A135473 n=3: abc
%e A135473 n=4: aabc, abbc, abcc
%e A135473 n=5: aaabc, abbbc, abccc, aabbc, aabcc, abbcc, ababc, abcbc
%Y A135473 Cf. A135017, A135156, A135157, A135475, A135479, A130838.
%Y A135473 Cf. also A137739, A137740, A137741, A137742, A137743, A137744, A137745, A137746, A137747, A137748.
%K A135473 nonn
%O A135473 1,4
%A A135473 _Max Alekseyev_, Jan 07 2008
%E A135473 a(19) - a(33) from _David Applegate_, Feb 12 2008
%E A135473 Extended to 37 terms by _David Applegate_, Feb 16 2008
%E A135473 Thanks to Robert Mercas and others for comments and references - _N. J. A. Sloane_, Apr 26 2013