OFFSET
1,2
COMMENTS
a(n) - a(n-1) is in {1,2,3} for n>=2, and a(n)/n -> 4 - sqrt(5).
From Michel Dekking, Aug 29 2020: (Start)
This sequence is a generalized Beatty sequence.
Recall from A286749 that A286749 is the letter-to-letter image of the fixed point x of the morphism mu given by
mu: 1->12341, 2->1, 3->2, 4->34.
where the letter-to-letter map lambda is defined by
lambda: 1->1, 2->1, 3->0, 4->0.
The return words of the word 1 in x are A:=1 and B:=1234.
We have mu(1)=12341, and mu(1234)=123411234. So the derived morphism is tau: A->BA, B->BAB.
This morphism happens to be the square of the Fibonacci morphism on the alphabet {B,A}.
The return word A = 1 has lambda-image 1, and the return word B = 1234 has lambda-image 1100. This means that they give distances 1, respectively 1 and 3 between (successive) occurrences of 1's in A286749. This leads to the decoration B->13, A->1, which amounts to replacing 0 by 01 and 1 by 0 in the Fibonacci word.
But the Fibonacci word is fixed by the {0,1} version of tau. It follows that the sequence (a(n+1)-a(n)) = 1,3,1,1,3,... is the Fibonacci word on the alphabet {1,3}. Finally, Lemma 8 in the paper "Generalized Beatty sequences..." then gives that a(n) = 5*n-2*floor(n*phi)-2, for n>0.
Clearly this implies a(n)/n -> 4 - sqrt(5).
(End)
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018-2019.
FORMULA
a(n) = 5*n-2*floor(n*phi)-2. - Michel Dekking, Aug 29 2020
EXAMPLE
As a word, A286749 = 11001110011001110011010..., in which 1 is in positions 1,2,5,6,7,10,...
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 14 2017
STATUS
approved