A305392 - OEIS

%I #27 Mar 07 2020 13:50:20

%S 2,1,2,1,2,1,2,2,1,2,1,2,1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1,2,1,2,2,1,2,

%T 1,2,1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1,2,1,2,1,2,2,1,2,1,

%U 2,1,2,1,2,1,2,2,1,2,1,2,1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1

%N First differences of A140100.

%C a(n) seems to take only the values 1 or 2, where {a(n), a(n+1)} may be {2, 1} or {1, 2} or {2, 2}, but not {1, 1}, and where {a(n), a(n+1), a(n+2), a(n+3)} may be {2, 1, 2, 1} or {1, 2, 1, 2} or {1, 2, 2, 1}, but not {2, 1, 1, 2}. The second differences of A140100 (first differences of this sequence) thus seem to take only the values -1 or 0 or 1. - _Daniel Forgues_, Aug 17 2018

%C From _Michel Dekking_, Mar 16 2019: (Start)

%C Let x be the tribonacci word x = A092782 = 1,2,1,3,1,2,1,1,...

%C Consider the morphism delta:

%C       1 -> 2212121212121,

%C       2 -> 22121212121,

%C       3 -> 2212121.

%C Conjecture: (a(n)) =  212121 delta(x).

%C (End)

%C Conjecture: This sequence (prefixed by 1 since A140100 should really begin with 0) is 1.TTW(2,1,1) where TTW is the ternary tribonacci word defined in A080843,  or equally it is THETA(2,1,1), where THETA is defined in A275925. - _N. J. A. Sloane_, Mar 19 2019

%C All these conjectures are now theorems - see the Dekking et al. paper. - _N. J. A. Sloane_, Jul 22 2019

%H N. J. A. Sloane, <a href="/A305392/b305392.txt">Table of n, a(n) for n = 1..49999</a>

%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.

%F a(n) = A140100(n+1)-A140100(n).

%Y For first differences of A140100, A140101, A140102, A140103 see A305392, A305374, A305393, A305394.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jun 23 2018