

A014677


First differences of A001468.


1



1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1
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OFFSET

0,1


COMMENTS

A001468 is an infinite Fibonacci word with strings of 2's of length A001468(n) delimited by 1's.  Paul D. Hanna, Dec 17 2004
c(n):=a(n1), n >= 1, is 1 if n is a Wythoff Bnumber from A001950, it is 0 if n=A(B(m)+1) for some m >= 1, with A(k):=A000201(k) (Wythoff Anumbers) and it is +1 if n=A(A(m)+1)=B(m)+1 for some m >= 0, with B(0):=0.  Wolfdieter Lang, Oct 13 2006
This sequence is a symbolic sequence as discussed in the paper "Morphisms, Symbolic Sequences, and Their Standard Forms". It can be derived directly from the 2block morphism induced by the morphism generating A001468. Since A001468 is the Fibonacci word A003849, but on the alphabet {2,1}, with an extra 1 in front, this 2block morphism has 3symbol Fibonacci as a fixed point: A270788. The 2blocks in A001468 are 12, 21, and 22, yielding the differences a(n) = 1, a(n) = 1, and a(n) = 0. In 3symbol Fibonacci these correspond to the letters 2, 1, and 3. Expressing this coding with pi given by pi(1)=1, pi(2)=1, pi(3)=0, we obtain the formula below. Wolfdieter Lang's Wythoff description of (a(n)) follows from the corresponding Wythoff description in A270788.  Michel Dekking, Dec 30 2019


LINKS

Table of n, a(n) for n=0..91.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.


FORMULA

abs(a(n)) = floor(f*ceiling(n/f))  ceiling(f*floor(n/f)) where f=phi=(1+sqrt(5))/2; for n > 1, abs(a(n)) = A005713(n1).  Benoit Cloitre, Apr 21 2003
G.f. equals the continued fraction: A(x) = [0;1, 1/x, 1/x, 1/x^2, 1/x^3, 1/x^5, 1/x^8, ..., 1/x^Fibonacci(n), ...].  Paul D. Hanna, Dec 17 2004
a(n) = b(n)  b(n1) with b(n):=A005614(n), n >= 1.
a(n) = pi(A270788(n)), n >= 1, where pi is the lettertoletter map pi(1)=1, pi(2)=1, pi(3)=0.  Michel Dekking, Dec 30 2019


CROSSREFS

Cf. A001468, A000045. Essentially equal to A270788.
Sequence in context: A325321 A255887 A295316 * A307425 A210826 A307421
Adjacent sequences: A014674 A014675 A014676 * A014678 A014679 A014680


KEYWORD

sign


AUTHOR

N. J. A. Sloane, Nov 07 2001


STATUS

approved



