OFFSET
1,1
COMMENTS
The sequence of first differences of A003146 (the third of the trio A003144, A003145, A003146) is equal to A276788 + A276789 + 1.
Also first differences of A278040.- Wolfdieter Lang, Dec 05 2018
From Michel Dekking, Mar 21 2019: (Start)
(a(n)) is a fixed point of the tribonacci morphism on the alphabet {4,3,2}, i.e., the morphism given by 4 -> 43, 3 -> 42, 2 -> 4.
To see this, let U := baca, V := baa, W := ba be the three return words of the letter b in the tribonacci word
x = abacabaabacaba... = aUVUW...
[See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]
Under the tribonacci morphism tau given by
tau(a) = ab, tau(b) = ac, tau(c) = a
one obtains
tau(U) = acabaab = b^{-1} UV b,
tau(V) = acabab = b^{-1} UW b,
tau(W) = acab = b^{-1} U b,
which is conjugate to the tribonacci morphism on the alphabet {U,V,W}.
Since these words have lengths 4, 3, and 2, the result follows.
(End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10608
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See page 317.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Jacques Justin and Laurent Vuillon, Return words in Sturmian and episturmian words, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
FORMULA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 14 2016
STATUS
approved