A090822 -id:A090822 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 05 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 >= 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

			From _M. F. Hasler_, Aug 09 2018: (Start)
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)

Dion Gijswijt, Table of n, a(n) for n = 1..2000
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.
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.
Index entries for sequences related to Gijswijt's sequence

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

Cf. A091787 for the concatenation of the glue strings.

# 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

A091411 Lengths of the B blocks in analysis of A090822.

Original entry on oeis.org

1, 3, 9, 19, 47, 98, 220, 441, 885, 1771, 3551, 7106, 14279, 28559, 57121, 114243, 228495, 456994, 914012, 1828025, 3656053, 7312107, 14624223, 29248450, 58497096, 116994195, 233988391, 467976791, 935953586, 1871907196

Offset: 1

N. J. A. Sloane, Mar 04 2004

nonn

Also, values of len_y(n) when len_x(n) = 0 in A090822.

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.
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 [pdf, ps].
Levi van de Pol, The Growth Rate of Gijswijt's Sequence, J. Int. Seq. (2025) Vol. 28, Art. No. 25.4.6.
Index entries for sequences related to Gijswijt's sequence

Cf. A090822, A091579, A091410, A095828.

a(1) = 1; for n > 1, a(n+1) = 2*a(n) + A091579(n).

This roughly doubles at each step and a(n) -> 1.743349432191828... * 2^n.

14279 and 28559 from Allan Wilks, Mar 04 2004

Extended by N. J. A. Sloane, Mar 06 2004

A093955 A090822(n) - 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 0

Offset: 1

N. J. A. Sloane, May 22 2004

nonn

The purpose of A093955-A093958 is to compare A090822, A091787, A091799 and A091844 on the same "scale".

Sequence is unbounded. A 3 appears for the first time at n=220. A 4 appears for the first time at around n = 10^10^23.

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. A090822, A093956-A093958.

A157644 Indices of 3's in A090822.

Original entry on oeis.org

9, 18, 28, 37, 41, 45, 46, 56, 65, 75, 84, 88, 92, 93, 97, 107, 116, 126, 135, 139, 143, 144, 154, 163, 173, 182, 186, 190, 191, 195, 199, 203, 204, 208, 212, 216, 217, 218, 219, 229, 238, 248, 257, 261, 265, 266, 276, 285, 295, 304, 308, 312, 313, 317, 327, 336, 346, 355

Offset: 1

Paul Curtz, Mar 03 2009

nonn

There is a group a(i+39) = a(i)+220 for i=1,..,39, then a group a(i+39)=a(i)+220 for i=40,...,78; these correlations become irregular later in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A157334, A107223, A157645.

Clear[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]; Position[Table[a[n], {n, 1, 355}], 3] // Flatten (* Jean-François Alcover, Mar 25 2013 *)

Extended by R. J. Mathar, Mar 15 2009

A091409 a(n) is the smallest m such that A090822(m) = n.

Original entry on oeis.org

1, 3, 9, 220

Offset: 1

N. J. A. Sloane, based on a suggestion from Dion Gijswijt (gijswijt(AT)science.uva.nl), Mar 04 2004

nonn

Dion C. Gijswijt, Krulgetallen, Pythagoras, 55ste Jaargang, Nummer 3, Jan 2016. (Shows that the sequence is infinite)
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.
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 [pdf, ps].
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.
Index entries for sequences related to Gijswijt's sequence

Cf. A090822.

a(n) is about 2^(2^(3^(4^(5^...^(n-1))))).

Sequence is infinite but next term, about 10^(10^23.09987) (see A091787), is too large to include.

A157107 Indices of 4's in A090822.

Original entry on oeis.org

220, 440, 661, 881, 1105, 1325, 1546, 1766, 1991, 2211, 2432, 2652, 2876, 3096, 3317, 3537, 3771, 3991, 4212, 4432, 4656, 4876, 5097, 5317, 5542, 5762, 5983, 6203, 6427, 6647, 6868, 7088, 7326, 7546, 7767, 7987, 8211, 8431, 8652, 8872, 9097, 9317, 9538

Offset: 1

Paul Curtz, Feb 23 2009

nonn

Despite appearances, this is a highly irregular sequence. - N. J. A. Sloane
The first pair of consecutive 4's occurs at position 255895648634818208370064452304769558261700170817472823398081\
655524438021806620809813295008281436789493636145 [Gijswijt, 2016, page 12]. - Sergio Pimentel, Feb 21 2017
Comment from N. J. A. Sloane, Feb 28 2017: (Start)
Page 12 of Gijswijt (2016) states:
"In dat stuk vinden we 355 promoties en daarmee 355 stukken lijm van lengte 1, 3, 1, 9, 4, 24, 1, 3, ... Na 355 keer verdubbelen en lijm toevoegen vinden we uiteindelijk de eerste dubbele vier in A090822, en wel op positie
255.895.648.634.818.208.370.064.452.304.769.558.
261.700.170.817.472.823.398.081.655.524.438.021.
806.620.809.813.295.008.281.436.789.493.636.145".
A rough translation might be:
"... we find 355 promotions and thus 355 glue segments of lengths 1, 3, 1, 9, 4, 24, 1, 3, ...
We eventually find the first pair of consecutive 4's in A090822 at position
  255,895,648,634,818,208,370,064,452,304,769,558,
  261,700,170,817,472,823,398,081,655,524,438,021,
  806,620,809,813,295,008,281,436,789,493,636,145."
Dion Gijswijt, Mar 01 2017 adds that the number mentioned is the position of the second 4 in the first occurrence of 4,4 in A090822. For the glue segments see A091579.
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..300
Dion Gijswijt, Krulgetallen, Pythagoras, 55ste Jaargang, Nummer 3, Jan 2016.

Cf. A090822, A091579.

Edited by N. J. A. Sloane, Feb 24 2009

More terms from R. J. Mathar, Feb 24 2009

A094781 Array T(i,j), i>=1, j >= 1, forming a two-dimensional version of A090822, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 12 2004

T(1,i) = T(i,1) = A090822(i). For i and j > 1, T(i,j) = max {k1, k2}, where k1 = curling number of (T(i,1), T(i,2)...,T(i,j-1)), k2 = curling number of (T(1,j), T(2,j)...,T(i-1,j)).

The curling number of a finite string S = (s(1),...,s(n)) is the largest integer k such that S can be written as xy^k for strings x and y (where y has positive length).

			Array begins:
1 1 2 1 1 2 2 2 3 1 1 2 1 1 2 2 2 3 2 1 ... (A090822)
1 1 2 1 1 2 2 2 3 1 1 2 1 1 2 2 2 3 2 1 ... (A090822)
2 2 2 3 2 2 2 3 2 2 2 3 3 2 ... (A091787)
1 1 3 1 1 3 3 2 1 1 2 1 1 2 ... (A094782)
1 1 2 1 1 2 2 2 3 1 2 1 1 2 ... (A094839)
2 2 2 3 2 1 1 2 1 2 3 2 2 3 ...
2 2 2 3 2 1 1 3 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, 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. A090822, A091787, A094782.

A156799 Indices of 1's in Gijswijt's sequence A090822.

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 13, 14, 20, 21, 23, 24, 29, 30, 32, 33, 48, 49, 51, 52, 57, 58, 60, 61, 67, 68, 70, 71, 76, 77, 79, 80, 99, 100, 102, 103, 108, 109, 111, 112, 118, 119, 121, 122, 127, 128, 130, 131, 146, 147, 149, 150, 155, 156, 158, 159, 165, 166, 168, 169, 174

Offset: 1

Paul Curtz, Feb 16 2009

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Gijswijt's sequence

Increased all values by 1, because A090822 has offset 1. R. J. Mathar, Feb 25 2009

A157041 Indices of 2's in A090822.

Original entry on oeis.org

3, 6, 7, 8, 12, 15, 16, 17, 19, 22, 25, 26, 27, 31, 34, 35, 36, 38, 39, 40, 42, 43, 44, 47, 50, 53, 54, 55, 59, 62, 63, 64, 66, 69, 72, 73, 74, 78, 81, 82, 83, 85, 86, 87, 89, 90, 91, 94, 95, 96, 98, 101, 104, 105, 106, 110, 113, 114, 115, 117, 120, 123, 124, 125, 129, 132

Offset: 1

Paul Curtz, Feb 22 2009

nonn

Despite appearances, this is a highly irregular sequence. - N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Edited by N. J. A. Sloane, Feb 24 2009

Incremented all values by 1 to match the offset in A090822. R. J. Mathar, Feb 24 2009

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

Original entry on oeis.org

1, 2, 3, 7, 19, 39, 95, 197, 441, 883, 1771, 3543, 7103, 14213, 28559, 57119

Offset: 1

N. J. A. Sloane, Mar 04 2004

So far this very nearly doubles at each step.

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. A090822, A091411.

28559 and 57119 from Allan Wilks, Mar 04 2004

cp's OEIS Frontend

A091579 Lengths of suffix blocks associated with A090822.

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A091411 Lengths of the B blocks in analysis of A090822.

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A093955 A090822(n) - 1.

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A157644 Indices of 3's in A090822.

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A091409 a(n) is the smallest m such that A090822(m) = n.

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A157107 Indices of 4's in A090822.

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A094781 Array T(i,j), i>=1, j >= 1, forming a two-dimensional version of A090822, read by antidiagonals.

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A156799 Indices of 1's in Gijswijt's sequence A090822.

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A157041 Indices of 2's in A090822.

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

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