A091411 -id:A091411 - cp's OEIS frontend

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

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".

A157140 A091411(k+3) replicated 2^k times, k= 0, 1, 2, 3, ...

Original entry on oeis.org

9, 19, 19, 47, 47, 47, 47, 98, 98, 98, 98, 98, 98, 98, 98, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 220, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441, 441

Offset: 0

Paul Curtz, Feb 24 2009

nonn

Generated from the quadrisection b(n) =  4, 13, 23, 32, 51, 60, 70, 79, 102, 111, 121, 130,... of A156799 as follows:
1) b(n+1)-b(n) = 9, 10, 9, 19, 9, 10, 9, 23, 9, 10, 9, 19,..,
2) b(n+2)-b(n) = 19, 19, 28, 28, 19, 19, 32, 32, 19, 19, 28, 28,...,
3) b(n+4)-b(n) = 47, 47, 47, 47, 51, 51, 51, 51, 47, 47, 47, 47,...,
4) b(n+8)-b(n) = 98, 98, 98, 98, 98, 98, 98, 98, 122, 122, 122, 122, 122, 122, 122, 122, 98, .. .
The sum of neighbors at the first jump of b(n+2^k) is 9+10 = 19 (k=0), 19+28 = 47 (k=1), 47+51 = 98 (k=2), 98+122 = 220 (k=3), and in general A091411(k+3).

A157217 Mix A091411 and its differences.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 19, 28, 47, 51, 98, 122, 220, 221, 441, 444, 885, 886, 1771, 1780, 3551, 3555, 7106, 7173, 14279, 14280, 28559, 28562, 57121, 57122, 114243, 114252, 228495, 228499, 456994, 457018, 914012, 914013, 1828025, 1828028, 3656053, 3656054, 7312107, 7312116

Offset: 0

Paul Curtz, Feb 25 2009

First differences: 1, 1, 3, 3, 1, 9, 9, 19, 4, 47, ....

a(2n) = A091441(n+1). a(2n-1) = A091441(n+1)-A091441(n).

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.

A090822 Gijswijt's sequence: a(1) = 1; for n>1, a(n) = largest integer k such that the word a(1)a(2)...a(n-1) is of the form xy^k for words x and y (where y has positive length), i.e., the maximal number of repeating blocks at the end of the sequence so far.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 1

Offset: 1

Dion Gijswijt, Feb 27 2004

Here xy^k means the concatenation of the words x and k copies of y.
The name "Gijswijt's sequence" is due to N. J. A. Sloane, not the author!
Fix n and suppose a(n) = k. Let len_y(n) = length of shortest y for this k and let len_x = n-1 - k*len_y(n) = corresponding length of x. A091407 and A091408 give len_y and len_x. For the subsequence when len_x = 0 see A091410 and A091411.
The first 4 occurs at a(220) (see A091409).
The first 5 appears around term 10^(10^23).
We believe that for all N >= 6, the first time N appears is at about position 2^(2^(3^(4^(5^...^(N-1))))). - N. J. A. Sloane and Allan Wilks, Mar 14 2004
For a similar formula, see p. 6 of Levi van de Pol article. - Levi van de Pol, Feb 06 2023
In the first 100000 terms the fraction of [1's, 2's, 3's, 4's] seems to converge, to about [.287, .530, .179, .005] respectively. - Allan Wilks, Mar 04 2004
When k=12 is reached, say, it is treated as the number 12, not as 1,2. This is not a base-dependent sequence.
Does this sequence have a finite average? Does anyone know the exact value? - Franklin T. Adams-Watters, Jan 23 2008
Answer: Given that "...the fraction of [1's, 2's, 3's, 4's] seems to converge, to about [.287, .530, .179, .005]..." that average should be the dot product of these vectors, i.e., about 1.904. - M. F. Hasler, Jan 24 2008
Second answer: The asymptotic densities of the numbers exist, and the average is the dot product. See pp. 56-59 of Levi van de Pol article. - Levi van de Pol, Feb 06 2023
Which is the first step with two consecutive 4's? Or the shortest run found so far between two 4's? - Sergio Pimentel, Oct 10 2016
Answer: The first x such that the x-th and (x+1)-th element are 4, is 255895648634818208370064452304769558261700170817472823... ...398081655524438021806620809813295008281436789493636144. See p. 55 of Levi van de Pol article. - Levi van de Pol, Feb 06 2023

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.

N. J. A. Sloane, Table of n, a(n) for n = 1..24000
Andreas Abel and Andres Loeh, Haskell program
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.
Benjamin Chaffin and N. J. A. Sloane, The Curling Number Conjecture, preprint.
Dion C. Gijswijt, Krulgetallen, Pythagoras, 55ste Jaargang, Nummer 3, Jan 2016, (Article in Dutch about this sequence, see pages 10-13, cover and back cover).
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. 2.
N. J. A. Sloane, Seven Staggering Sequences.
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
Index entries for sequences related to Gijswijt's sequence
Index entries for sequences related to curling numbers

Cf. A091407, A091408, A091409 (records), A091410, A091411, A091579, A091586, A091970, A093955-A093958.
A091412 gives lengths of runs. A091413 gives partial sums.
Cf. also A094004, A160766, A217206, A217207.
Generalizations: A094781, A091975, A091976, A092331-A092335.

Haskell
```
-- See link.
```

K:= proc(L)
local n,m,k,i,b;
m:= 0;
n:= nops(L);
for k from 1 do
  if k*(m+1) > n then return(m) fi;
  b:= L[-k..-1];
  for i from 1 while i*k <= n and L[-i*k .. -(i-1)*k-1] = b do od:
  m:= max(m, i-1);
od:
end proc:
A[1]:= 1:
for i from 2 to 220 do
  A[i]:= K([seq(A[j],j=1..i-1)])
od:
seq(A[i],i=1..220); # Robert Israel, Jul 02 2015

ClearAll[a]; reversed = {a[2]=1, a[1]=1}; blocs[len_] := Module[{bloc1, par, pos}, bloc1 = Take[reversed, len]; par = Partition[ reversed, len]; pos = Position[par, bloc_ /; bloc != bloc1, 1, 1]; If[pos == {}, Length[par], pos[[1, 1]] - 1]]; a[n_] := a[n] = Module[{an}, an = Table[{blocs[len], len}, {len, 1, Quotient[n-1, 2]}] // Sort // Last // First; PrependTo[ reversed, an]; an]; A090822 = Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Aug 13 2012 *)

A090822(n,A=[])={while(#Ak||break; k=m);A=concat(A,k));A} \\ M. F. Hasler, Aug 08 2018

def k(s):
    maxk = 1
    for m in range(1, len(s)+1):
        i, y, kk = 1, s[-m:], len(s)//m
        if kk <= maxk: return maxk
        while s[-(i+1)*m:-i*m] == y: i += 1
        maxk = max(maxk, i)
def aupton(terms):
    alst = [1]
    for n in range(2, terms+1):
        alst.append(k(alst))
    return alst
print(aupton(99)) # Michael S. Branicky, Mar 28 2022

cp's OEIS Frontend

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

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A157140 A091411(k+3) replicated 2^k times, k= 0, 1, 2, 3, ...

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A157217 Mix A091411 and its differences.

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

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A090822 Gijswijt's sequence: a(1) = 1; for n>1, a(n) = largest integer k such that the word a(1)a(2)...a(n-1) is of the form xy^k for words x and y (where y has positive length), i.e., the maximal number of repeating blocks at the end of the sequence so far.

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A091410 Values of n such that len_x(n) = 0 in A090822.

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A095828 Smoothed lengths of the B blocks in analysis of A090822.

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