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A091579 Lengths of suffix blocks associated with A090822.

%I A091579 #42 Aug 19 2025 09:35:29
%S A091579 1,3,1,9,4,24,1,3,1,9,4,67,1,3,1,9,4,24,1,3,1,9,4,196,3,1,9,4,24,1,3,
%T A091579 1,9,4,68,3,1,9,4,24,1,3,1,9,4,581,3,1,9,4,25,3,1,9,4,67,1,3,1,9,4,24,
%U A091579 1,3,1,9,4,196,3,1,9,4,24,1,3,1,9,4,68,3,1,9,4,24,1,3,1,9,4,1731,3,1,9,4,24
%N A091579 Lengths of suffix blocks associated with A090822.
%C A091579 The suffix blocks are what is called "glue string" in the paper by Gijswijt et al (2007). Roughly speaking, these are the terms >= 2 appended before the sequence (A090822) goes on with a(n+1) = 1 followed by all other initial terms a(2..n), cf. Example. The concatenation of these glue strings yields A091787. - _M. F. Hasler_, Aug 08 2018
%H A091579 Dion Gijswijt, <a href="/A091579/b091579.txt">Table of n, a(n) for n = 1..2000</a>
%H A091579 Fokko J. van de Bult, Dion C. Gijswijt, John P. Linderman, N. J. A. Sloane, and Allan R. Wilks, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Sloane/sloane55.html">A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
%H A091579 Levi van de Pol, <a href="https://arxiv.org/abs/2209.04657">The first occurrence of a number in Gijswijt's sequence</a>, arXiv:2209.04657 [math.CO], 2022.
%H A091579 Levi van de Pol, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Vandepol/vandepol5.html">The Growth Rate of Gijswijt's Sequence</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.4.6.
%H A091579 <a href="/index/Ge#Gijswijt">Index entries for sequences related to Gijswijt's sequence</a>
%e A091579 From _M. F. Hasler_, Aug 09 2018: (Start)
%e A091579 In sequence A090822, after the initial (1, 1) follows the first suffix block or glue string (2) of length a(1) = 1. This is followed by A090822(4) = 1 which indicates that the suffix block has ended, and the whole sequence A090822(1..3) up to and including this suffix block is repeated: A090822(4..6) = A090822(1..3).
%e A091579 Then A090822 goes on with (2, 2, 3, 1, ...), which tells that the second suffix block is A090822(7..9) = (2, 2, 3) of length a(2) = 3, whereafter the sequence starts over again: A090822(10..18) = A090822(1..9). (End)
%o A091579 (Python)
%o A091579 # compute curling number of L
%o A091579 def curl(L):
%o A091579     n = len(L)
%o A091579     m = 1 #max nr. of repetitions at the end
%o A091579     k = 1 #length of repeating block
%o A091579     while(k*(m+1) <= n):
%o A091579         good = True
%o A091579         i = 1
%o A091579         while(i <= k and good):
%o A091579             for t in range(1, m+1):
%o A091579                 if L[-i-t*k] != L[-i]:
%o A091579                     good = False
%o A091579             i = i+1
%o A091579         if good:
%o A091579             m = m+1
%o A091579         else:
%o A091579             k = k+1
%o A091579     return m
%o A091579 # compute lengths of first n glue strings
%o A091579 def A091579_list(n):
%o A091579     Promote = [1] #Keep track of promoted elements
%o A091579     L = [2]
%o A091579     while len(Promote) <= n:
%o A091579         c = curl(L)
%o A091579         if c < 2:
%o A091579             Promote = Promote+[len(L)+1]
%o A091579             c = 2
%o A091579         L = L+[c]
%o A091579     return [Promote[i+1]-Promote[i] for i in range(n)]
%o A091579 # _Dion Gijswijt_, Oct 08 2015
%Y A091579 Cf. A090822, A091587 (records). For a smoothed version see A091839.
%Y A091579 Cf. A091787 for the concatenation of the glue strings.
%K A091579 nonn
%O A091579 1,2
%A A091579 _N. J. A. Sloane_, Mar 05 2004