A123564 - OEIS

%I #15 Mar 09 2022 01:48:58

%S 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,

%T 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

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

%C 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) )

%C Essentially equal to A143667. - _Michel Dekking_, Sep 26 2017

%H G. C. Greubel, <a href="/A123564/b123564.txt">Table of n, a(n) for n = 1..5000</a>

%H Michel Dekking and Mike Keane, <a href="https://arxiv.org/abs/2202.13548">Two-block substitutions and morphic words</a>, arXiv:2202.13548 [math.CO], 2022.

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

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

%e a(1) = 2*1+0 = 2;

%e a(2) = 2*1+1 = 3;

%e a(3) = 2*0+1 = 1.

%t 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 *)

%o (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

%Y Cf. A005614, A143667

%K easy,nonn

%O 1,1

%A _Alexandre Losev_, Nov 12 2006