Levi van de Pol - cp's OEIS frontend

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

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".

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".

A357065 Numbers k with the following property: the value A091839(k+1) is not a 1 that is obtained from smoothing A091579.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 63, 65, 66, 67, 69, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 93, 95, 97, 98, 99, 101, 103

Offset: 1

Levi van de Pol, Sep 10 2022

This sequence is the function iota1 in the article "The first occurrence of a number in Gijswijt's sequence" (page 21). For the connection with smoothing, see Subsection 8.1.

			14 is not a term since A091839(14+1) is a 1 obtained from smoothing: in A091579, the eleventh value is 4, which is replaced by 3,1 to obtain the fourteenth and fifteenth terms of A091839.

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), Article 07.1.2.
Levi van de Pol, The first occurrence of a number in Gijswijt's sequence, arXiv:2209.04657 [math.CO], 2022.

Cf. A091787, A090822, A357066, A357067.

A357063 Lengths of the B blocks associated with A091787.

Original entry on oeis.org

1, 4, 13, 42, 127, 382, 1149, 3448, 10345, 31044, 93133, 279400, 838203, 2514610, 7543831, 22631496, 67894489, 203683468, 611050413, 1833151240, 5499453721, 16498361166, 49495083499, 148485250498, 445455751497, 1336367254492, 4009101763477, 12027305290463, 36081915871390, 108245747614173, 324737242842520, 974211728527561, 2922635185582686, 8767905556748059, 26303716670244178, 78911150010732543, 236733450032197630, 710200350096592891

Offset: 1

Levi van de Pol, Sep 10 2022

nonn

The B blocks are explained in the article "A slow-growing sequence defined by an unusual occurrence". They have superscript (2).

			The third B-block of order 2 is B_3^{(2)}=2223222322233. Therefore, a(3)=13.

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), Article 07.1.2.

Cf. A091787, A357068.

number_of_terms=38
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
# This algorithm generates a prefix of the level-3 Gijswijt sequence
def Generate_A3(number):
  glue_lengths=[]
  A3=[3]
  S=[3]
  i=0
  while(True):
      c=Cn(A3)
      if c<3:
        glue_lengths.append(len(S))
        i=i+1
        if i==number:
            break
        S=[]
      A3.append(max(c,3))
      S.append(max(c,3))
  return glue_lengths
glue_lengths=Generate_A3(number_of_terms-1)
beta_lengths=[1]
beta_length=1
for l in glue_lengths:
    beta_length=3*beta_length+l
    beta_lengths.append(beta_length)
print(beta_lengths)

By special permission, more than the usual number of terms are shown. - N. J. A. Sloane, Oct 23 2022

A357066 Decimal expansion of the limit of k/A357065(k) as k goes to infinity.

Original entry on oeis.org

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

Offset: 0

Levi van de Pol, Oct 21 2022

In the article "The first occurrence of a number in Gijswijt's sequence", this constant is called nu_1. Its existence is proven in Proposition 6.13(c). 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.

			0.69167220878112615338...

Levi van de Pol, The first occurrence of a number in Gijswijt's sequence, arXiv:2209.04657 [math.CO], 2022.
Levi van de Pol, Gijswijt-sequences: Code for article on Gijswijt sequences, zenodo. (Algorithm nu_m.py)

Cf. A090822, A091409, A357065.

Python
```
# See nu_m.py program in zenodo link
```

A357064 a(n) = k such that A091411(k) = A091409(n).

Original entry on oeis.org

1, 2, 3, 7, 418090195952691922788354

Offset: 1

Levi van de Pol, Sep 10 2022

nonn

The existence of a(n) is proven in Lemma 1.2(a) of the article "The first occurrence of a number in Gijswijt's sequence". There, it is called t^{(1)}(n). In this article, a formula for the numbers t^{(m)}(n) is given. It looks like a tower of exponents and can be found in Theorem 6.20. This formula is then used to find a formula for the first occurrence of an integer n in Gijswijt's sequence, which is A091409(n).
The value of a(5) is calculated in Subsection 8.2 of the same article.
The value of a(6) is larger than 10^(10^100), so it would be impossible to include here.

			For n=4 we have A091411(7)=A091409(4). Therefore, a(4)=7.

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. 7.

Cf. A091409, A091411.

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User: Levi van de Pol

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

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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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A357065 Numbers k with the following property: the value A091839(k+1) is not a 1 that is obtained from smoothing A091579.

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A357063 Lengths of the B blocks associated with A091787.

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A357066 Decimal expansion of the limit of k/A357065(k) as k goes to infinity.

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A357064 a(n) = k such that A091411(k) = A091409(n).

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