%I #46 Mar 07 2020 13:50:20

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

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

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

%N First differences of A003144.

%C In A276790, leave 2's unchanged, but replace 1's by 2's and 0's by 1's, and then omit the initial 1.

%C If we prefixed A003144 with an initial 0, then its first differences would be a' := 1 followed by a, that is, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, ... If we now add 1 to every term of a' we get A305374 = first differences of A140101. - _N. J. A. Sloane_, Jul 17 2018

%C This relation between A003144 and A140101 is a conjecture - _Michel Dekking_, Mar 18 2019 [It has been a theorem since Mar 22 2019. - _N. J. A. Sloane_, Jun 25 2019. (See the Dekking et al. paper)]

%C (a(n)) is a morphic sequence: in the tribonacci word A092782 = 1,2,1,3,1,2,1,1,... map 1 -> 2, 2 -> 2, 3 -> 1. - _Michel Dekking_, Mar 21 2019

%H Robert Israel, <a href="/A276788/b276788.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 316.

%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) = A003144(n+1) - A003144(n), n >= 1.

%F a(n+1) = 2 - t(n)*(t(n) - 1)/2 = 2 - A276791(n+1), for n >= 0, where t(n) = A080843(n). See the W. Lang link in A080843, eq. (38). - _Wolfdieter Lang_, Dec 06 2018

%p M:= 10: # to use M generations of strings

%p S[1]:="a": S[2]:="ab": S[3]:="abac":

%p for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:

%p P:=select(t -> S[M][t]="a", [$1..length(S[M])]):

%p P[2..-1]-P[1..-2]; # _Robert Israel_, Nov 01 2016

%Y Cf. A003144, A003145, A003146, A080843, A276790, A276791, A275925, A305374, A140101.

%Y See A278039 for partial sums.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Oct 14 2016