A135473 - OEIS

A135473

a(n) = number of strings of length n that can be obtained by starting with abc and repeatedly doubling any substring in place.

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

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OFFSET

1,4

COMMENTS

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.

REFERENCES

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.

LINKS

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

Index entries for doubling substrings

FORMULA

Empirically, grows like 3^n.

EXAMPLE

n=3: abc

n=4: aabc, abbc, abcc

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

CROSSREFS