A091799 -id:A091799 - cp's OEIS frontend

A091842 Lengths of suffix blocks associated with A091799.

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

A093957 A091799(n) - 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 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.

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, A093955-A093958.

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

A091844 a(1) = 4. 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,4).

Original entry on oeis.org

4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5

Offset: 1

N. J. A. Sloane, Mar 10 2004

nonn

Here xy^k means the concatenation of the words x and k copies of y.

The first '6' occurs at a(3908). - Sergio Pimentel, Jul 13 2015

Giovanni Resta, Table of n, a(n) for n = 1..10000
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, A091799, A091845.

maxBlockLength = 100; a[1] = 4; a[n_] := a[n] = Module[{rev = Reverse[Array[a, n - 1]]}, blockCount[blockLength_] := Module[{par, p1, k}, par = Partition[rev, blockLength]; If[par == {}, Return[1]]; p1 = First[par]; k = 1; While[k <= Length[par], If[par[[k]] != p1, Break[], k++]]; k - 1]; Max[Max[Array[blockCount, maxBlockLength]], 4]]; Array[a, 100] (* Michael De Vlieger, Jul 13 2015, after Jean-François Alcover at A091799 *)

{A091844(Nmax, L=1, A=List(), f(A,m=3,L=0)=while( #A>=(m+1)*L++, while( A[#A-L*m+1..#A]==A[#A-L*(m+1)+1..#A-L], (m+++1)*L>#A&& break)); m,t) = while(#AA091799. - M. F. Hasler, Sep 30 2018

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.

A093958 A091844(n) - 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1

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.

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, A093955-A093958.

A093149 a(1) = 4; a(n) = (n^(n+1)+2*n-3)/(n-1) for n > 1.

Original entry on oeis.org

4, 9, 42, 343, 3908, 55989, 960802, 19173963, 435848052, 11111111113, 313842837674, 9726655034463, 328114698808276, 11966776581370173, 469172025408063618, 19676527011956855059, 878942778254232811940, 41660902667961039785745, 2088331858752553232964202

Offset: 1

N. J. A. Sloane, May 21 2004

nonn

This sequence represents the first step in which the Gijswijt's sequence (A090822), with a minimum value of n-1, reaches the value n+1 for the first time. For example, the first '3' in A090822 is in step 9, the first '4' in A091787 is in step 42, the first '5' in A091799 is in step 343 and so on. - Sergio Pimentel, Jul 15 2015

Cf. A090822, A091787, A091799, A091844.

[4] cat [(n^(n+1)+2*n-3)/(n-1): n in [2..20]]; // Vincenzo Librandi, Jul 16 2015

A093149:=n->(n^(n+1)+2*n-3)/(n-1): (4,seq(A093149(n), n=2..30)); # Wesley Ivan Hurt, Jul 15 2015

{4}~Join~Table[(n^(n + 1) + 2 n - 3)/(n - 1), {n, 2, 19}] (* Michael De Vlieger, Jul 13 2015 *)

main(size)=my(v=vector(size),i); v[1]=4; for(i=2,size, v[i]=(i^(i+1)+2*i-3)/(i-1)); v \\ Anders Hellström, Jul 13 2015

A093956 A091787(n) - 2.

Original entry on oeis.org

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

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, A093955-A093958.

cp's OEIS Frontend

A091842 Lengths of suffix blocks associated with A091799.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A093957 A091799(n) - 3.

Views

Author

Keywords

Comments

Links

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Python

A091844 a(1) = 4. 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,4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A093955 A090822(n) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

A093958 A091844(n) - 4.

Views

Author

Keywords

Comments

Links

Crossrefs

A093149 a(1) = 4; a(n) = (n^(n+1)+2*n-3)/(n-1) for n > 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A093956 A091787(n) - 2.

Views

Author

Keywords

Comments

Links

Crossrefs