A091579 Lengths of suffix blocks associated with A090822.

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, 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, 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

Offset

1,2

Comments

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

Links

Dion Gijswijt, Table of n, a(n) for n = 1..2000

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.

Levi van de Pol, The first occurrence of a number in Gijswijt's sequence, arXiv:2209.04657 [math.CO], 2022.

Index entries for sequences related to Gijswijt's sequence

Example

From M. F. Hasler, Aug 09 2018:
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).
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)

Prog

(Python)
# compute curling number of L
def curl(L):
    n = len(L)
    m = 1 #max nr. of repetitions at the end
    k = 1 #length of repeating block
    while(k*(m+1) <= n):
        good = True
        i = 1
        while(i <= k and good):
            for t in range(1, m+1):
                if L[-i-t*k] != L[-i]:
                    good = False
            i = i+1
        if good:
            m = m+1
        else:
            k = k+1
    return m
# compute lengths of first n glue strings
def A091579_list(n):
    Promote = [1] #Keep track of promoted elements
    L = [2]
    while len(Promote) <= n:
        c = curl(L)
        if c < 2:
            Promote = Promote+[len(L)+1]
            c = 2
        L = L+[c]
    return [Promote[i+1]-Promote[i] for i in range(n)]
# Dion Gijswijt, Oct 08 2015

Crossrefs

Cf. A090822, A091587 (records). For a smoothed version see A091839.

Cf. A091787 for the concatenation of the glue strings.

Sequence in context: A124573 A127550 A021317 * A136159 A371746 A005533

Adjacent sequences: A091576 A091577 A091578 * A091580 A091581 A091582

Keyword

nonn

Author

N. J. A. Sloane, Mar 05 2004

Status

approved