OFFSET
1,1
COMMENTS
a(n) - a(n-1) is in {1,2,3,4} for n>=2, and a(n)/n -> (7 + sqrt(5))/4.
From Michel Dekking, Aug 29 2020: (Start)
There is an explicit expression for the difference sequence Delta a given by Delta a(n) = a(n+1)-a(n).
Claim: the sequence Delta a is the decoration a -> 14, b -> 13 of the Fibonacci word abaababaab......
Proof: 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.
We see from this that 0's in A286749 correspond uniquely to pairs 34 in x. So we compute the return words of 34. These are 3412 and 34112. Since
mu(3412) = 234123411, mu(34112) = 23412341123411,
the return words, coded as A = 3412, B = 34112 induce a descendant morphism
A->AB, B->ABB.
This well-known morphism (see A096270) has the property that its unique fixed point is the Fibonacci word (on the alphabet {B,A}), preceded by the letter A.
The return word 3412 has lambda-image 0011, and the return word 34112 has lambda-image 00111. This means that they give distances 1 and 3, respectively 1 and 4 between (successive) occurrences of 0's in A286749.
This leads to the decoration A->13, B->14.
That a(n)/n -> (7 + sqrt(5))/4 follows from the corresponding result for the sequence A286751.
(End)
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
EXAMPLE
As a word, A286749 = 11001110011001110011010..., in which 0 is in positions 3,4,8,9,12,...
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 14 2017
STATUS
approved