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A294180 The 3-symbol Pell word. 6
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)).

(a(n)) is the unique fixed point of the 3-symbol Pell morphism

    1 -> 123, 2 ->123, 3 -> 1.

The morphism and the fixed point are in standard form.

Modulo a change of alphabet (1->0, 2->1, 3->2), this sequence is equal to A263844.

From Michel Dekking, Feb 23 2018: (Start)

The positions of 1 in (a(n)) are given by

  A188376 = 1,4,7,8,11,14,15,18,...

Why is this true? First, the Pell word b is given by

    b(n) = [(n+1)(1-r)]-[n(1-r)], where r =1/sqrt(2).

This can rewritten as

    b(n) = [nr]-[(n+1)r]+1.

Second,

    1 occurs at n in (a(n))  <=>

    00 occurs at n in (b(n)) <=>

    b(n)+b(n+1) = 0          <=>

    [nr]-[(n+2)r]+2 = 0      <=>

    [(n+2)r]-[nr]-1 = 1      <=>

    1 occurs at n in A188374.

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)).

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)).

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)

See the comments of A188376 for a proof of the observation on the positions of 1 in (a(n)). - Michel Dekking, Feb 27 2018

LINKS

Michel Dekking, Table of n, a(n) for n = 1..1000

L. Carlitz, R. Scoville and V. E. Hoggatt, Jr.,Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

FORMULA

a(n) = floor((n+2)r)+floor((n+1)r)-2*floor(nr)+1, where r = 1-1/sqrt(2).

MAPLE

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)];

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A270788, A263844, A188376.

Sequence in context: A053839 A047896 A073645 * A179542 A082846 A117373

Adjacent sequences:  A294177 A294178 A294179 * A294181 A294182 A294183

KEYWORD

nonn

AUTHOR

Michel Dekking, Feb 11 2018

STATUS

approved

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Last modified October 16 01:14 EDT 2019. Contains 328038 sequences. (Running on oeis4.)