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A176075 Conway variant a(n) = 1+a(n-1-(n mod a(n-1))), with a(1)=1. 3
1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 10, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 12, 13, 12, 13, 14, 13, 14, 13, 14, 15, 13, 14, 15, 13, 14, 15, 14, 15, 14, 15, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence grows slowly. (a <= 100 for n <=3069, a <=200 for n<=12139, a<=300 for n <= 29003.)

Increasing long periods of repetition of the same number are interleaved by perturbations (see the picture in the links section).

The first occurrence of jumps of magnitude |a(n+1)-a(n)|=k are:

k=2 -> n=24: |a(25)-a(24)|=|7-9|=2

k=3 -> n=147: |a(148)-a(147)|=|19-22|=3

k=4 -> n=152: |a(153)-a(152)|=|23-19|=4

k=5 -> n=560: |a(561)-a(560)|=|45-40|=5

k=6 -> n=12139: |a(12140)-a(12139)|=|194-200|=6

Bootstrapping from a(1)=2 would generate A000027 (starting from 2).

A similar sequence a(n)=1+a(1+(n mod a(n-1))), with a(1)=1 eventually enters the periodic sequence 3,2,3,4,3,3,4,2,3,4,4,2. With a(1)=2, the period is 60.

REFERENCES

G. Balzarotti and P. P. Lava, 103 curiosità matematiche, Hoepli, 2010, p. 274.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Paolo P. Lava, Plot of the first 10000 terms of the sequence

John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.

Klaus Pinn, A Chaotic Cousin Of Conway's Recursive Sequence, arXiv:cond-mat/9808031, 1998.

EXAMPLE

a(1)=1.

a(2)=1+a(2-1-(2 mod 1))=1+a(1-0)=1+a(1)=2.

a(3)=1+a(3-1-(3 mod 2))=1+a(2-1)=1+a(1)=2.

a(4)=1+a(4-1-(4 mod 2))=1+a(3-0)=1+a(3)=3.

a(5)=1+a(5-1-(5 mod 3))=1+a(4-2)=1+a(2)=3.

a(6)=1+a(6-1-(6 mod 3))=1+a(5-0)=1+a(5)=4.

a(7)=1+a(7-1-(7 mod 4))=1+a(6-3)=1+a(3)=3.

MAPLE

P:=proc(i) local a, n; a:=array(1..100000); a[1]:=1; print(a[1]); for n from 2 by 1 to i do a[n]:=1+a[n-1-(n mod a[n-1])]; print(a[n]); od; end: P(100000);

# alternative program

A176075 := proc(n) option remember; if n = 1 then 1 else 1+procname(n-1-(n mod procname(n-1))) ;  end if; end proc: # R. J. Mathar, Jan 23 2011

MATHEMATICA

a[1] = 1; a[n_] := a[n] = 1 + a[n - 1 - Mod[n, a[n - 1]]];

Array[a, 80] (* Jean-François Alcover, Dec 13 2017 *)

CROSSREFS

Cf. A004001, A176076.

Sequence in context: A070241 A066412 A196048 * A256555 A117119 A208280

Adjacent sequences:  A176072 A176073 A176074 * A176076 A176077 A176078

KEYWORD

nonn

AUTHOR

Paolo P. Lava and Giorgio Balzarotti, Apr 09 2010

STATUS

approved

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Last modified September 30 12:21 EDT 2020. Contains 337439 sequences. (Running on oeis4.)