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

Author

John W. Layman, Apr 04 2011

Keywords

• nonn

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.

Crossrefs

Cf. A003589, A003849, A006190.

Programs

Mathematica

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

Formula

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