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
KEYWORD
nonn
AUTHOR
Paolo P. Lava and Giorgio Balzarotti, Apr 09 2010
STATUS
approved