Alexandre Losev - cp's OEIS frontend

A178992 Ordered list in decimal notation of the subwords (with leading zeros omitted) appearing in the infinite Fibonacci word A005614 (0->1 & 1->10).

0, 1, 2, 3, 5, 6, 10, 11, 13, 21, 22, 26, 27, 43, 45, 53, 54, 86, 90, 91, 107, 109, 173, 181, 182, 214, 218, 346, 347, 363, 365, 429, 437, 693, 694, 726, 730, 858, 859, 875, 1387, 1389, 1453, 1461, 1717, 1718, 1750, 2774, 2778, 2906, 2907, 2923, 3435, 3437, 3501

Offset: 1

Alexandre Losev, Jan 03 2011

The definition mentions the Fibonacci word A005614. Note that the official Fibonacci word is A003849, which would give a different list, namely, the 2's-complement of the present list. - N. J. A. Sloane, Jan 12 2011

			The Fibonacci word has a minimal complexity, i.e., for any n there are n+1 distinct subwords of length n (see for example Allouche and Shallit).
E.g. for n=1 they are '0' and '1', for n=2 '01', '10' and '11' or, in decimal notation '1','2',and '3'.
Some subwords prefixed with '0' have the same decimal value as shorter ones, but there is no real ambiguity as double zeros do not appear in the infinite Fibonacci word.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

T. D. Noe, Table of n, a(n) for n = 1..1015
T. D. Noe, Table of n, a(n) for n = 1..1652 [The first 1652 terms written in binary, including leading zeros. Not a b-file.]

Cf. A003849, A005614, A171676, A179969.

iter=8; f=Nest[Flatten[# /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, iter]; u={}; n=1; While[lst={}; k=0; While[num=FromDigits[Take[f, {1, n}+k], 2]; lst=Union[lst, {num}]; Length[lst]

Definition clarified by N. J. A. Sloane, Jan 10 2011

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

Original entry on oeis.org

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

Offset: 1

Alexandre Losev, Nov 12 2006

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

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

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.

Cf. A005614, A143667

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 *)

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

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.

A112539 Half-baked Thue-Morse: at successive steps the sequence or its bit-inverted form is appended to itself.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0

Offset: 1

Alexandre Losev, Dec 15 2005

			Triangle begins:
  1;
  0;
  1, 0;
  0, 1, 0, 1;
  1, 0, 1, 0, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0;
  1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0;
  ...

Cf. A112865 (as +-1), A341389 (complement).

Cf. A104104, A010060.

s = {1}; Do[s = Join[s, Mod[s + 1, 2]]; s = Join[s, s], {n, 4}]; s (* Robert G. Wilson v, Dec 22 2005 *)

aiter(x) = my(s=[1]); for(i=1, x, s=concat(s, if(i%2, [1-e|e<-s], s))); s \\ Ruud H.G. van Tol, Jan 20 2025

s = [1]
for _ in range(4):
    s = s + [(x + 1) % 2 for x in s]
    s = s + s
print(s) # Robert C. Lyons, Jan 19 2025

More terms from Robert G. Wilson v, Dec 22 2005

A108882 Period doubling sequence starting with '1 0 1'.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1

Offset: 1

Alexandre Losev, Jul 14 2005

Start with S = 1,0,1; replace S by SS and complement the last bit; iterate.

Similar to A035263 but with a different start.

Cf. A035263.

f[l_] := Block[{s = Flatten[{l, l}]}, s[[ -1]] = Mod[s[[ -1]] + 1, 2]; s]; Nest[f, {1, 0, 1}, 6] (* Robert G. Wilson v, Jul 16 2005 *)

def f(l):
    s = l + l
    s[-1] = (s[-1] + 1) % 2
    return s
result = [1, 0, 1]
for _ in range(6):
    result = f(result)
print(result) # Robert C. Lyons, Jan 19 2025

More terms from Robert G. Wilson v, Jul 16 2005

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User: Alexandre Losev

A178992 Ordered list in decimal notation of the subwords (with leading zeros omitted) appearing in the infinite Fibonacci word A005614 (0->1 & 1->10).

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

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A112539 Half-baked Thue-Morse: at successive steps the sequence or its bit-inverted form is appended to itself.

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A108882 Period doubling sequence starting with '1 0 1'.

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