A091840 -id:A091840 - cp's OEIS frontend

A091841 Records in A091840.

Original entry on oeis.org

1, 3, 9, 32, 119, 463, 1837, 7332, 29307, 117203, 468785

Offset: 0

N. J. A. Sloane, Mar 10 2004

Each term is roughly 4 times the previous term.

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.
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 [pdf, ps].
Index entries for sequences related to Gijswijt's sequence

Cf. A091787, A090822, A091579, A091840.

a(8)-a(10) from John P. Linderman, May 30 2004

A091787 a(1) = 2. To get a(n+1), write the string a(1)a(2)...a(n) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence so far. Then a(n+1) = max(k,2).

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2

Offset: 1

N. J. A. Sloane, Mar 07 2004

nonn

Here xy^k means the concatenation of the words x and k copies of y.
a(77709404388415370160829246932345692180) = 5 is the first time 5 appears.
This is also the concatenation of the glue strings of A090822, whose respective lengths are given in A091579. - M. F. Hasler, Oct 04 2018
This sequence is called the level-2 Gijswijt sequence.

			To get a(2): a(1) = 2 = (2)^1, so k = 1, a(2) = 2.
To get a(3): a(1)a(2) = 22 = (2)^2, so a(3) = k = 2.
To get a(4): a(1)a(2)a(3) = 222 = (2)^3, so a(3) = k = 3.

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Fokko J. van de Bult, Dion C. Gijswijt, John P. Linderman, N. J. A. Sloane, and Allan R. Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
Benjamin Chaffin, John P. Linderman, N. J. A. Sloane, and Allan R. Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102 [math.CO], Dec 25 2012.
Benjamin Chaffin, John P. Linderman, N. J. A. Sloane, and Allan R. Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
N. J. A. Sloane, Seven Staggering Sequences.
Levi van de Pol, The first occurrence of a number in Gijswijt's sequence, arXiv:2209.04657 [math.CO], 2022.
Levi van de Pol, The Growth Rate of Gijswijt's Sequence, J. Int. Seq. (2025) Vol. 28, Art. No. 25.4.6. See p. 4.
Index entries for sequences related to Gijswijt's sequence

Cf. A090822, A091799, A091840.

A091787(n, A=[])={while(#Ak||break; k=m); A=concat(A, k)); A} \\ M. F. Hasler, Oct 04 2018

from itertools import islice
def c(w):
    for k in range(len(w), 0, -1):
        for l in range(1, len(w)//k + 1):
            if w[-k*l:] == w[-l:]*k: return k
def agen(): # generator of terms
    alst, an = [], 2
    while True: yield an; alst.append(an); an = max(2, c(alst))
print(list(islice(agen(), 99))) # Michael S. Branicky, Sep 10 2022

A091588 A smoothed version of A091587.

Original entry on oeis.org

1, 3, 8, 24, 67, 195, 580, 1730, 5179, 15533, 46578, 139712, 419115, 1257319, 3771930, 11315764, 33947261, 101841751, 305525228, 916575642, 2749726883

Offset: 0

N. J. A. Sloane, Mar 05 2004

Each term is roughly 3 times the previous term.

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.
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 [pdf, ps].
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

Records in A091839. Cf. A090822, A091587, A091587, A217590.

a(n+1) = a(n) + A357063(n+1) + A091840(n+1). See Conjecture 4.2 of F. J. van de Bult et al., proved p. 54 of Levi van de Pol. - Levi van de Pol, Nov 04 2022

a(10)-a(13) from Allan Wilks, Mar 10 2004
a(14)-a(20) from Alexander Staunton, Apr 09 2022
Removed an incorrect program. - N. J. A. Sloane, Aug 20 2022

A091842 Lengths of suffix blocks associated with A091799.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 10, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 10, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 10, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 42, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 10, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 10

Offset: 1

N. J. A. Sloane, Mar 10 2004

nonn

The suffix blocks are what is called "glue string" in the paper by Gijswijt et al. (2007). Roughly speaking, these are the terms appended before the sequence goes on with a copy of all its initial terms up to the current position. (In the present sequence, when this happens, then this initial segment will actually be repeated for a total of 4 copies. Therefore each suffix block will start with a "4".) - M. F. Hasler, Aug 08 2018

			The first suffix block or "glue string" of length 10 is "4454444455", occurring as A091799(5760309077..5760309086). This is also the first occurrence of "55" in A091799. The first suffix block of length 42 is "4454444455"."444445"^5."55" (where . is concatenation) which occurs approximately at position 4.56*10^38. This is also the first occurrence of "555" in A091799. - _M. F. Hasler_, Aug 08 2018, corrected Sep 30 2018

M. F. Hasler, Table of n, a(n) for n = 1..1500
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.
Index entries for sequences related to Gijswijt's sequence

Cf. A091787, A090822, A091579, A091840, A091843, A091799.

print_A091842(LIM=oo,A=[],c=#A)={while(#Ak||break; k=m); A=concat(A,max(k,4)); if(k<4,#A>1&&print1(#A-c",");c=#A))} \\ M. F. Hasler, Aug 09 2018

a(n) = A091843(valuation(n-1,4)) for n < 259. For larger n, the index n must be increased by the number of terms "200" which occur* up to n-1 (* e.g., at n = 256, 511, 766, 1277, 1532, ...). - M. F. Hasler, Aug 09 2018

A091843 Records in A091842.

Original entry on oeis.org

1, 3, 10, 42, 200, 983, 4892, 24434, 122141

Offset: 0

N. J. A. Sloane, Mar 10 2004

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.
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 [pdf, ps].
Index entries for sequences related to Gijswijt's sequence

Cf. A091787, A090822, A091579, A091840.

a(6)-a(8) from John P. Linderman, May 30 2004

A357068 Decimal expansion of the limit of A357063(k)/3^(k-1) as k goes to infinity.

Original entry on oeis.org

1, 5, 7, 7, 2, 2, 7, 9, 2, 3, 9, 9, 4, 5, 0, 0, 6, 9, 4, 1, 0

Offset: 1

Levi van de Pol, Oct 24 2022

In the article "The first occurrence of a number in Gijswijt's sequence", this constant is called epsilon_2. Its existence is proved in Theorem 7.2. The constant occurs in a direct formula (Theorem 7.11) for the first occurrence of an integer n in the level-2 Gijswijt sequence A091787.

			1.57722792399450069410...

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

Cf. A091787, A357063, A091840, A357066, A357067.

Equal to 1 + Sum_{k>=1} A091840(k)/3^k. Proved in Corollary 7.3 of the article "The first occurrence of a number in Gijswijt's sequence".

A091845 Records in sequence of lengths of suffix blocks associated with A091844.

Original entry on oeis.org

1, 3, 11, 55, 315, 1872, 11205, 67195

Offset: 0

N. J. A. Sloane, Mar 10 2004

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.
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 [pdf, ps].
Index entries for sequences related to Gijswijt's sequence

Cf. A091787, A090822, A091579, A091840.

a(5)-a(7) from John P. Linderman, May 30 2004.

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A091841 Records in A091840.

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A091787 a(1) = 2. To get a(n+1), write the string a(1)a(2)...a(n) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence so far. Then a(n+1) = max(k,2).

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A091588 A smoothed version of A091587.

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A091842 Lengths of suffix blocks associated with A091799.

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A091843 Records in A091842.

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A357068 Decimal expansion of the limit of A357063(k)/3^(k-1) as k goes to infinity.

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A091845 Records in sequence of lengths of suffix blocks associated with A091844.

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