|
%I #28 Oct 06 2018 04:30:11
%S 1,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,
%T 2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,
%U 2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,1,2,2,1,2,2,2,1,2
%N First differences of Beatty sequence A022838(n) = floor(n sqrt(3)).
%C Let S(0) = 1; obtain S(k) from S(k-1) by applying 1 -> 221, 2 -> 2221; sequence is S(0), S(1), S(2), ... - _Matthew Vandermast_, Mar 25 2003
%C The sequence (a(n+1)-1) is the homogeneous Sturmian sequence with slope sqrt(3)-1, which is fixed point of the morphism 0->110, 1->1101. So (a(n), n>0) is the unique fixed point of the morphism 1->221, 2->2212. - _Michel Dekking_, Oct 06 2018
%C The dual version defined by b(n)=1-(a(n)-1) for n>0 is the Sturmian sequence with slope 1-(sqrt(3)-1) = 2-sqrt(3). It is the fixed point of the morphism 0->0010, 1->001. The sequence (b(n)) prefixed with 0 equals A275855. - _Michel Dekking_, Oct 06 2018
%H Michel Dekking, <a href="http://math.colgate.edu/~integers/sjs7/sjs7.Abstract.html">Substitution invariant Sturmian words and binary trees</a>, Integers 18A, #A7, 1-15 (2018).
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F a(n) = A007538(n+1) - 1. - _Reinhard Zumkeller_, Feb 13 2012
%t Flatten[ NestList[ Flatten[ # /. {1 -> {2, 2, 1}, 2 -> {2, 2, 2, 1}}] &, {1}, 4]] (* _Robert G. Wilson v_, Jun 20 2005 *)
%t Differences[Floor[Range[0,110]Sqrt[3]]] (* _Harvey P. Dale_, Mar 15 2018 *)
%o (Haskell)
%o a080757 = (subtract 1) . a007538 . (+ 1)
%o -- _Reinhard Zumkeller_, Feb 14 2012
%Y Equals A007538(n+1) - 1. Cf. A001030.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Mar 25 2003
|
