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%I #40 Mar 07 2020 13:52:25
%S 4,3,4,2,4,3,4,4,3,4,2,4,3,4,3,4,2,4,3,4,4,3,4,2,4,3,4,2,4,3,4,4,3,4,
%T 2,4,3,4,3,4,2,4,3,4,4,3,4,2,4,3,4,4,3,4,2,4,3,4,3,4,2,4,3,4,4,3,4,2,
%U 4,3,4,2,4,3,4,4,3,4,2,4,3,4,3,4,2,4,3,4,4,3,4,2,4,3,4,3,4,2,4,3
%N First differences of A003145.
%C The sequence of first differences of A003146 (the third of the trio A003144, A003145, A003146) is equal to A276788 + A276789 + 1.
%C Also first differences of A278040.- _Wolfdieter Lang_, Dec 05 2018
%C From _Michel Dekking_, Mar 21 2019: (Start)
%C (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.
%C To see this, let U := baca, V := baa, W := ba be the three return words of the letter b in the tribonacci word
%C x = abacabaabacaba... = aUVUW...
%C [See Justin & Vuillon (2000) for definition of return word. - _N. J. A. Sloane_, Sep 23 2019]
%C Under the tribonacci morphism tau given by
%C tau(a) = ab, tau(b) = ac, tau(c) = a
%C one obtains
%C tau(U) = acabaab = b^{-1} UV b,
%C tau(V) = acabab = b^{-1} UW b,
%C tau(W) = acab = b^{-1} U b,
%C which is conjugate to the tribonacci morphism on the alphabet {U,V,W}.
%C Since these words have lengths 4, 3, and 2, the result follows.
%C (End)
%H Robert Israel, <a href="/A276789/b276789.txt">Table of n, a(n) for n = 1..10608</a>
%H Elena Barcucci, Luc Belanger and Srecko Brlek, <a href="http://www.fq.math.ca/Papers1/42-4/quartbarcucci04_2004.pdf">On tribonacci sequences</a>, Fib. Q., 42 (2004), 314-320. See page 317.
%H F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i1p52/8039">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H Jacques Justin and Laurent Vuillon, <a href="http://www.numdam.org/item/ITA_2000__34_5_343_0/">Return words in Sturmian and episturmian words</a>, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%F a(n) = A003145(n+1) - A003145(n) = A278040(n) - A278040(n-1) = 4 - A080843(n-1), for n >= 1. See eq. (20) of the W. Lang link. - _Wolfdieter Lang_, Dec 04 2018
%Y Cf. A003144, A003145, A003146, A080843, A276788, A278040.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Oct 14 2016
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