A082215 -id:A082215 - cp's OEIS frontend

A123121 Length of the n-th Zimin word (A082215(n)).

1, 3, 7, 15, 31, 63, 127, 255, 511, 1024, 2050, 4102, 8206, 16414, 32830, 65662, 131326, 262654, 525310, 1050622, 2101246, 4202494, 8404990, 16809982, 33619966, 67239934, 134479870, 268959742, 537919486, 1075838974, 2151677950, 4303355902, 8606711806

Offset: 1

Dmitry Kamenetsky, Sep 29 2006

The length of the n-th Zimin word on a countably infinite alphabet {x_1, x_2, x_3, ...} with Z_{n+1} = Z_n x_{n+1} Z_n (as opposed to the use of base 10 in A082215) is 2^n-1. - Danny Rorabaugh, Mar 12 2015

			The Zimin words are defined by Z_1 = 1, Z_n = Z_{n-1}nZ_{n-1}.
So the Zimin words are 1, 121, 1213121, 121312141213121 ...

M. Lothaire, Algebraic combinatorics on words, Cambridge University Press, Cambridge, 2002.

Jiří Balun, Tomáš Masopust, and Petr Osička, Speed Me up if You Can: Conditional Lower Bounds on Opacity Verification, arXiv:2304.09920 [cs.FL], 2023.
J. Cooper and D. Rorabaugh, Bounds on Zimin Word Avoidance, arXiv:1409.3080 [math.CO], 2014; Congressus Numerantium, 222 (2014), 87-95.
L. J. Cummings and M. Mays, A one-sided Zimin construction, Electron. J. Combin. 8 (2001), #R27.
A. I. Zimin, Blocking sets of terms, Math. USSR Sbornik, 47 (1984), No. 2, 353-364.

Cf. A082215.

[n le 1 select 1 else 2*Self(n-1) + Ceiling(Log(n+1)/Log(10)): n in [1..40]]; // Vincenzo Librandi, Sep 26 2015

A[1]:= 1:
for i from 2 to 100 do A[i]:= 2*A[i-1]+ilog10(i+1) od:
seq(A[i],i=1..100); # Robert Israel, Sep 18 2014

a(n) = 2*a(n-1) + ceiling(log_10(n+1)).

G.f.: sum(j>=1, x^(10^j))/(1-3*x+2*x^2). - Robert Israel, Sep 18 2014

More terms from Vincenzo Librandi, Sep 26 2015

A018238 Add 1 to leading digit and put in front.

Original entry on oeis.org

1, 21, 3121, 41213121, 5121312141213121, 61213121412131215121312141213121, 7121312141213121512131214121312161213121412131215121312141213121

Offset: 1

N. J. A. Sloane, Michael Minic (minic(AT)mtsu.edu)

The concatenation of first n terms (if n is small) yields a palindrome: 1, 121, 1213121, etc. - Amarnath Murthy, Apr 08 2003
From M. F. Hasler, May 05 2008: (Start)
This is not the case from n=10 on: According to the formula in A123121 A082215(10) has an even number of digits, the middle digits being "10". (In a strict sense, e.g. Def. 3 of the first reference there, A082215(9) is the last Zimin word on the alphabet {1,...,9}, though.)
While there is less ambiguity about the definition of A018238(10), it is not clear if A018238(11) should start with "11..." or with "10..." (the largest digit of all subsequent terms being "9"). According to the formula in A123121, a(100) has 3 digits more than a(99), so the first choice seems appropriate and has been adopted for the given PARI code.
However, it corresponds to a modified definition, "a(n) = concatenation of n and all preceding terms". a(3) is the only prime term up to a(14) included. The sequence is (1,0,1,0,1,0,...) (mod 3), at least up to a(20). (End)

M. F. Hasler, Table of n, a(n) for n=1,...,11.

Cf. A001511, A082215, A123121, A033564.

A018238(n,t="")=for(k=2,n,t=Str(t,k-1,t));eval(Str(n,t)) \\ M. F. Hasler, May 05 2008

cp's OEIS Frontend

A123121 Length of the n-th Zimin word (A082215(n)).

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