OFFSET
1,2
COMMENTS
Alternative descriptions (1): unique positive integer sequence taking values in {1,2} satisfying a(1)=1, a(2)=2 and a(a(1)+...+a(n))=a(n) for n >= 3.
(2) Start with 1,2; then for any k>=1, a(a(1)+...+a(k))=a(k), fill in any undefined terms by the rule that a(t) = 1 if a(t-1) = 2 and a(t) = 2 if a(t-1) = 1.
(3) a(1)= 1, a(2)=2, a(a(1)+a(2)+...+a(n))=a(n); a(a(1)+a(2)+...+a(n)+1)=3-a(n).
More generally, the sequence a(n)=floor(r*(n+2))-floor(r*(n+1)), r= (1/2) *(z+sqrt(z^2+4)), z integer >=1, is defined by a(1), a(2) and a(a(1)+a(2)+...+a(n)+f(z))=a(n); a(a(1)+a(2)+...+a(n)+f(z)+1)=(2z+1)-a(n) where f(1)=0, f(z)=z-2 for z>=2.
FORMULA
EXAMPLE
a(1)+a(2)=3 and a(a(1)+a(2)) must be a(2) so a(3)=2. Therefore a(a(1)+a(2)+a(3))=a(5)=2 and from the rule the "hole" a(4) is 1. Hence sequence begins 1,2,2,1,2,...
MAPLE
A082389:=n->floor((n+2)*(1+sqrt(5))/2) - floor((n+1)*(1+sqrt(5))/2): seq(A082389(n), n=1..300); # Wesley Ivan Hurt, Jan 16 2017
MATHEMATICA
Rest@Nest[ Flatten[ # /. {1 -> 2, 2 -> {2, 1}}] &, {1}, 11] (* Robert G. Wilson v, Jan 26 2006 *)
#[[2]]-#[[1]]&/@Partition[Table[Floor[GoldenRatio n], {n, 0, 110}], 2, 1] (* Harvey P. Dale, Sep 04 2019 *)
PROG
(Python)
from math import isqrt
def A082389(n): return (n+2+isqrt(m:=5*(n+2)**2)>>1)-(n+1+isqrt(m-10*n-15)>>1) # Chai Wah Wu, Aug 29 2022
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Benoit Cloitre, Apr 14 2003
STATUS
approved