A176075 - OEIS

%I #25 Dec 13 2017 11:50:51

%S 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,

%T 9,10,10,10,10,11,10,11,10,11,11,11,11,11,11,11,11,11,11,11,12,12,12,

%U 12,12,13,12,13,12,13,14,13,14,13,14,15,13,14,15,13,14,15,14,15,14,15,14

%N Conway variant a(n) = 1+a(n-1-(n mod a(n-1))), with a(1)=1.

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

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

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

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

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

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

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

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

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

%C 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.

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

%H Paolo P. Lava, <a href="/A176075/b176075.txt">Table of n, a(n) for n = 1..10000</a>

%H Paolo P. Lava, <a href="/A176075/a176075.pdf">Plot of the first 10000 terms of the sequence</a>

%H John A. Pelesko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pelesko/pel11.html">Generalizing the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.

%H Klaus Pinn, <a href="https://arxiv.org/abs/cond-mat/9808031">A Chaotic Cousin Of Conway's Recursive Sequence</a>, arXiv:cond-mat/9808031, 1998.

%e a(1)=1.

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

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

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

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

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

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

%p 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);

%p # alternative program

%p 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

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

%t Array[a, 80] (* _Jean-François Alcover_, Dec 13 2017 *)

%Y Cf. A004001, A176076.

%K nonn

%O 1,2

%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Apr 09 2010