A188590 [(n+1)*r] - [n*r], where r = 3/2 + sqrt(13)/2 and [...] denotes the floor function.

3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3

Offset

1,1

Comments

It appears that this sequence is a fixed-pt of the morphism 3 -> 334, 4 -> 3343, starting with 3. The orbit of 3 under the indicated morphism is 3, 334, 3343343343, 334334334333433433433343343343334, ...

The sequence of the lengths of the words in this orbit appears to be A006190 = {1,3,10,33,109,...}, a solution of the difference equation a(n) = 3*a(n-1) + a(n-2). A root of the auxiliary equation r^2 - 3r -1 = 0 of this difference equation is 3/2 + sqrt(13)/2, the value of r used in the definition of {a(n)}.

See A003849 for the infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit).

It appears that {a(n)-1} = {2,2,3,2,2,3,2,2,3,2,2,2,3,...} is the same as A003589 (the number of 2's between consecutive 3's in A003589 gives the original sequence). This has been verified up to 2000 terms.

Links

Table of n, a(n) for n=1..76.

Formula

a(n) = [(n+1)*r] - [n*r], where r = 3/2 + sqrt(13)/2 and [...] denotes the floor function.

Mathematica

r = 3/2 + Sqrt[13]/2; Table[Floor[(n + 1)r] - Floor[n * r], {n, 100}] (* Alonso del Arte, Apr 04 2011 *)

Crossrefs

Cf. A003589, A003849, A006190.

Sequence in context: A082978 A005536 A172515 * A080038 A121937 A003034

Adjacent sequences: A188587 A188588 A188589 * A188591 A188592 A188593

Keyword

nonn

Author

John W. Layman, Apr 04 2011

Status

approved