A091579 -id:A091579 - 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

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

A357067 Decimal expansion of the limit of A091411(k)/2^(k-1) as k goes to infinity.

Original entry on oeis.org

3, 4, 8, 6, 6, 9, 8, 8, 6, 4, 3, 8, 3, 6, 5, 5, 9, 7, 0, 2, 3, 5, 8, 7, 2, 7, 0, 0, 7, 0, 2, 2, 2, 0, 6, 6, 7, 3, 3, 5, 4, 1, 3, 6, 6, 2

Offset: 1

Levi van de Pol, Oct 22 2022

In the article "The first occurrence of a number in Gijswijt's sequence", this constant is called epsilon_1. Its existence is proved in Theorem 7.2. The constant occurs in a direct formula (Theorem 7.11) for A091409(n), the first occurrence of the integer n in Gijswijt's sequence A090822.

			3.48669886438365597023...

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

Cf. A090822, A091409, A091411, A091579, A357066, A357068.

import math
from mpmath import *
# warning: 0.1 and mpf(1/10) are incorrect. Use mpf(1)/mpf(10)
mp.dps=60
def Cn(X):
    l=len(X)
    cn=1
    for i in range(1, int(l/2)+1):
        j=i
        while(X[l-j-1]==X[l-j-1+i]):
            j=j+1
            if j>=l:
                break
        candidate=int(j/i)
        if candidate>cn:
            cn=candidate
    return cn
def epsilon():
    A=[2] # level-2 Gijswijt sequence
    number=1 # number of S strings encountered
    position=0 # position of end of last S
    value=mpf(1) # approximation for epsilon1
    for i in range(1,6000):
        k=Cn(A)
        A.append(max(2,k))
        if k<2:
            value=value+mpf(i-position)/mpf(2**number)
            position=mpf(i)
            number+=1
    return value
print("epsilon_1: ",epsilon())

Equal to 1 + Sum_{k>=1} A091579(k)/2^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.

cp's OEIS Frontend

A091841 Records in A091840.

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A091842 Lengths of suffix blocks associated with A091799.

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A091843 Records in A091842.

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A357067 Decimal expansion of the limit of A091411(k)/2^(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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