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

%I A123564 #19 Jun 08 2025 01:12:10
%S A123564 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 A123564 2,3,1,1,2,3,1,1,2,3,1,1,2,2,3,1,2,2,3,1,1,2,3,1,1,2,3,1,1,2,2,3,1,2,
%U A123564 2,3,1,1,2,3,1,1,2,3,1,1,2,2,3,1,2,2,3,1
%N A123564 The infinite Fibonacci word reencoded by writing successive non-overlapping pairs of bits as decimal numbers.
%C A123564 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 A123564 Essentially equal to A143667. - _Michel Dekking_, Sep 26 2017
%H A123564 G. C. Greubel, <a href="/A123564/b123564.txt">Table of n, a(n) for n = 1..5000</a>
%H A123564 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 A123564 f = (sqrt(5)-1)/2; m = 2*n; a(n) = floor(m*f) - 2*floor((m-1)*f) + floor((m+1)*f);
%F A123564 a(n) = 2*A005614(2n-1) + A005614(2n), using the infinite Fibonacci word A005614.
%e A123564 a(1) = 2*1+0 = 2;
%e A123564 a(2) = 2*1+1 = 3;
%e A123564 a(3) = 2*0+1 = 1.
%t A123564 f := 1/GoldenRatio; T[n_] := Floor[2*n*f] - 2*Floor[(2*n - 1)*f] + Floor[(2*n + 1)*f]; Table[T[n], {n, 100}] (* _G. C. Greubel_, Oct 16 2017 *)
%o A123564 (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 A123564 Cf. A005614, A143667
%K A123564 easy,nonn
%O A123564 1,1
%A A123564 _Alexandre Losev_, Nov 12 2006