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A123564 The infinite Fibonacci word reencoded by writing successive non-overlapping pairs of bits as decimal numbers. 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A308644 A308184 A114280 * A036466 A065882 A276327
KEYWORD
easy,nonn
AUTHOR
Alexandre Losev, Nov 12 2006
STATUS
approved

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Last modified April 18 16:22 EDT 2024. Contains 371780 sequences. (Running on oeis4.)