A123564 The infinite Fibonacci word reencoded by writing successive non-overlapping pairs of bits as decimal numbers.

2, 3, 1, 1, 2, 3, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 1, 2, 3, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2

Offset

1,1

Comments

The algorithm used here suggests multiple variations such as using more than 2 bits, allowing overlap of successive subwords, using other numbers for the encoding of subwords or using other binary sequences. (E.g. overlapping: a(n) = 2*A005614(n) + A005614(n+1) )

Essentially equal to A143667. - Michel Dekking, Sep 26 2017

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.

Formula

f = (sqrt(5)-1)/2; m = 2*n; a(n) = floor(m*f) - 2*floor((m-1)*f) + floor((m+1)*f);

a(n) = 2*A005614(2n-1) + A005614(2n), using the infinite Fibonacci word A005614.

Example

a(1) = 2*1+0 = 2;
a(2) = 2*1+1 = 3;
a(3) = 2*0+1 = 1.

Mathematica

f := 1/GoldenRatio; T[n_] := Floor[2*n*f] - 2*Floor[(2*n - 1)*f] + Floor[(2*n + 1)*f]; Transpose[{Range[1, 50], Table[T[n], {n, 1, 50}] (* G. C. Greubel, Oct 16 2017 *)

Prog

(PARI) f=(sqrt(5)-1)/2; a(n)= my(m=2*n); floor(m*f)-2*floor((m-1)*f)+floor((m+1)*f); \\ Michel Marcus, Sep 26 2017

Crossrefs

Cf. A005614, A143667

Sequence in context: A308644 A308184 A114280 * A036466 A065882 A276327

Adjacent sequences: A123561 A123562 A123563 * A123565 A123566 A123567

Keyword

easy,nonn

Author

Alexandre Losev, Nov 12 2006

Status

approved