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A294180 The 3-symbol Pell word.

%I A294180 #43 Jan 05 2025 19:51:41
%S A294180 1,2,3,1,2,3,1,1,2,3,1,2,3,1,1,2,3,1,2,3,1,2,3,1,1,2,3,1,2,3,1,1,2,3,
%T A294180 1,2,3,1,2,3,1,1,2,3,1,2,3,1,1,2,3,1,2,3,1,1,2,3,1,2,3,1,2,3,1,1,2,3,
%U A294180 1,2,3,1,1,2,3,1,2,3,1,2,3,1,1,2,3,1,2,3,1,1,2,3,1,2,3,1,1,2,3,1,2,3,1,2,3,1,1,2
%N A294180 The 3-symbol Pell word.
%C A294180 In the Pell word A171588 = 0, 0, 1, 0, 0, 1, 0, 0, 0, ..., group the letters in overlapping blocks of length two: [0,0],[0,1],[1,0],[0,0],[0,1],[1,0],... Then code [0,0]->1, [0,1]->2, [1,0]->3. This gives (a(n)).
%C A294180 (a(n)) is the unique fixed point of the 3-symbol Pell morphism
%C A294180     1 -> 123, 2 ->123, 3 -> 1.
%C A294180 The morphism and the fixed point are in standard form.
%C A294180 Modulo a change of alphabet (1->0, 2->1, 3->2), this sequence is equal to A263844.
%C A294180 From _Michel Dekking_, Feb 23 2018: (Start)
%C A294180 The positions of 1 in (a(n)) are given by
%C A294180   A188376 = 1,4,7,8,11,14,15,18,...
%C A294180 Why is this true? First, the Pell word b is given by
%C A294180     b(n) = [(n+1)(1-r)]-[n(1-r)], where r =1/sqrt(2).
%C A294180 This can rewritten as
%C A294180     b(n) = [nr]-[(n+1)r]+1.
%C A294180 Second,
%C A294180     1 occurs at n in (a(n))  <=>
%C A294180     00 occurs at n in (b(n)) <=>
%C A294180     b(n)+b(n+1) = 0          <=>
%C A294180     [nr]-[(n+2)r]+2 = 0      <=>
%C A294180     [(n+2)r]-[nr]-1 = 1      <=>
%C A294180     1 occurs at n in A188374.
%C A294180 The positions of 2 in (a(n)) are given by A001952 - 1 = 2,5,9,12,16,..., since 2 occurs at n in (a(n)) if and only if 3 occurs at n+1 in (a(n)).
%C A294180 The positions of 3 in (a(n)) are given by A001952 = 3,6,10,13,17,..., since 3 occurs at n in (a(n)) if and only if 1 occurs at n in (b(n)).
%C A294180 The sequence of positions of 3 in (a(n)) is equal to the sequence b in Carlitz et al. The sequence of positions of 1 in (a(n)) seems to be equal to the sequence ad' in Carlitz et al. (End)
%C A294180 See the comments of A188376 for a proof of the observation on the positions of 1 in (a(n)). - _Michel Dekking_, Feb 27 2018
%H A294180 Michel Dekking, <a href="/A294180/b294180.txt">Table of n, a(n) for n = 1..1000</a>
%H A294180 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr.,<a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian Representations</a>, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.
%H A294180 F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%F A294180 a(n) = floor((n+2)r)+floor((n+1)r)-2*floor(nr)+1, where r = 1-1/sqrt(2).
%p A294180 a:=[seq(floor((n+2)*(1-1/sqrt(2)))+floor((n+1)*(1-1/sqrt(2)))-2*floor(n*(1-1/sqrt(2)))+1, n=1..130)];
%t A294180 With[{r = 1 - 1/Sqrt[2]}, Table[Inner[Times, Map[Floor[(n + #) r] &, Range[0, 2]], {-2, 1, 1}, Plus] + 1, {n, 108}]] (* _Michael De Vlieger_, Feb 15 2018 *)
%o A294180 (Magma) [Floor((n+2)*r)+Floor((n+1)*r)-2*Floor(n*r)+1 where r is 1-1/Sqrt(2): n in [1..90]]; // _Vincenzo Librandi_, Feb 23 2018
%Y A294180 Cf. A270788, A263844, A188376.
%K A294180 nonn
%O A294180 1,2
%A A294180 _Michel Dekking_, Feb 11 2018