A005614 -id:A005614 - cp's OEIS frontend

A124841 Inverse binomial transform of A005614, the rabbit sequence: (1, 0, 1, 1, 0, ...).

1, -1, 2, -3, 3, 0, -10, 35, -90, 200, -400, 726, -1188, 1716, -2080, 1820, -312, -2704, 5408, 455, -39195, 170313, -523029, 1352078, -3114774, 6548074, -12668578, 22492886, -36020998, 49549110, -49549110, 0, 182029056, -670853984, 1809734560, -4242470755

Offset: 0

Gary W. Adamson, Nov 10 2006

sign

As with every inverse binomial transform, the numbers are given by starting from the sequence (A005614) and reading the leftmost values of the array of repeated differences.

			Given 1, 0, 1, 1, 0, ..., take finite difference rows:
1, 0, 1, 1, 0, ...
_-1, 1, 0, -1, ...
___ 2, -1, -1, ...
_____ -3, 0, ...
________ 3, ...
Left border becomes the sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A124842.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

A005614 = SubstitutionSystem[{0 -> {1}, 1 -> {1, 0}}, {1, 0}, 7] // Last;
Table[Differences[A005614, n], {n, 0, 35}][[All, 1]] (* Jean-François Alcover, Feb 06 2020 *)

Corrected and extended by R. J. Mathar, Nov 28 2011

A089910 Indices n at which blocks (1;1) occur in infinite Fibonacci word, i.e., such that A005614(n-1) = A005614(n-2) = 1.

Original entry on oeis.org

4, 9, 12, 17, 22, 25, 30, 33, 38, 43, 46, 51, 56, 59, 64, 67, 72, 77, 80, 85, 88, 93, 98, 101, 106, 111, 114, 119, 122, 127, 132, 135, 140, 145, 148, 153, 156, 161, 166, 169, 174, 177, 182, 187, 190, 195, 200, 203, 208, 211, 216, 221, 224, 229, 232, 237, 242, 245

Offset: 1

Benoit Cloitre, Nov 15 2003

nonn

a(n) is the number k such that floor(k/r) = floor(n*r^2), where r = golden ratio. - Clark Kimberling, May 03 2015

Clark Kimberling, Table of n, a(n) for n = 1..1000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathan Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823 [math.CO], 2014.
Urban Larsson and Nathan Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.

Cf. A000201, A001950, A026352, A270788.

phi:=(1+sqrt(5))/2:  seq(floor(phi*floor(n*phi^2))+1, n=1..80); # Michel Dekking, Sep 15 2016

r = GoldenRatio; u = Flatten[Table[Select[Range[Floor[(r^2 + r) n], Floor[(r^2 + r) n + 1]], Floor[#/r] == Floor[n*r^2] &], {n, 1, 100}]] (* Clark Kimberling, May 03 2015 *)

from math import isqrt
def A089910(n): return (n+isqrt(5*n**2)&-2)+n+1 # Chai Wah Wu, Aug 29 2022

a(n) = floor((2+sqrt(5))*n) + 0 or 1;
floor(n*(2+sqrt(5))) + b(a(n)) - a(n) = 0 where b(x) = A078588(x) = x + 1 + ceiling(x*sqrt(5)) - 2*ceiling(x*(1+sqrt(5))/2).
For n >= 2, a(n) = a(n-1) + d, where d = 5 if n-1 is in A000201, else d = 3. - Clark Kimberling, May 03 2015
a(n) = A003623(n) + 1 = A(B(n)) + 1, where A(B(n)) are the Wythoff AB-numbers. - Michel Dekking, Sep 15 2016

Definition corrected by Jeffrey Shallit, Dec 21 2023

A044432 a(n) is the number whose base-2 representation is d(0)d(1)...d(n), where d=A005614 (the infinite Fibonacci word).

Original entry on oeis.org

1, 2, 5, 11, 22, 45, 90, 181, 363, 726, 1453, 2907, 5814, 11629, 23258, 46517, 93035, 186070, 372141, 744282, 1488565, 2977131, 5954262, 11908525, 23817051, 47634102, 95268205, 190536410, 381072821, 762145643, 1524291286, 3048582573, 6097165147, 12194330294

Offset: 0

Clark Kimberling

a(n) can also be calculated as floor(2^n * R), where the rabbit constant R=0.709803442861291314641787399444575597012... converges rapidly using the result from Davison described in the comments at A014565. - Federico Provvedi, Oct 24 2018

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000225, A005614, A182028, A000079, A014565.

a044432 n = a044432_list !! n
a044432_list = scanl1 (\v b -> 2 * v + b) a005614_list
-- Reinhard Zumkeller, Apr 07 2012

FromDigits[(Floor[GoldenRatio(#+1)]-Floor[GoldenRatio #]-1)&@Range@#,2]&/@Range@40 (* Federico Provvedi, Oct 19 2018 *)
Floor[2^#/FromContinuedFraction[2^Fibonacci[Range[0,3*Max[1,Floor[2+Log[(#+1)/11]/ArcSinh[2]]]]]]]&/@Range[200] (* Federico Provvedi, Nov 01 2018 *)

a(n) = A000225(n+1) - A182028(n). - Reinhard Zumkeller, Apr 07 2012
a(n) = 2*a(n-1) + A005614(n) for n > 0, a(0) = 1. - Reinhard Zumkeller, Apr 07 2012
From Federico Provvedi, Oct 24 2018: (Start)
a(n) = A000079(n) * Sum_{k=0..n} ((floor(phi*(k+1)) - floor(phi*k) - 1)/2^k).
a(n) = floor(2^n*(1-Sum_{n >= 1} (-1)^(n+1)*(1+2^Fibonacci(3*n+1))/((2^(Fibonacci(3*n-1))-1)*(2^(Fibonacci(3*n + 2))-1)))).
a(n) = floor(2^n*R), where R is the rabbit constant.
a(n) = floor(2^n/[1, 2, 2, 4, 8, 32, ..., 2^Fibonacci(3*h)]), with h=1 for n=0, h=floor(2+log((n+1)/11)/arcsinh(2)) for n>0. (End)

Offset fixed by Reinhard Zumkeller, Apr 07 2012

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

Original entry on oeis.org

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

A028894 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +1 for 0, +3 for 1.

Original entry on oeis.org

1, 5, 23, 95, 381, 1527, 6109, 24439, 97759, 391037, 1564151, 6256607, 25026429, 100105719, 400422877, 1601691511, 6406766047, 25627064189, 102508256759, 410033027037, 1640132108151, 6560528432607, 26242113730429

Offset: 1

John McNamara, Jan 12 2002

nonn

Related to A005614, A002450, A037576, A066744.

f[1]={0}; f[2]={1}; f[n_] := f[n]=Join[f[n-1], f[n-2]]; a[1]=1; a[n_] := a[n]=4a[n-1]+1+2f[9][[n]]; a/@Range[1, 30]

Edited by Dean Hickerson, Jan 14 2002

A179969 Ordered list in decimal notation of the subwords appearing in the infinite Fibonacci word A005614.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 6, 5, 6, 10, 11, 13, 11, 13, 21, 22, 26, 27, 22, 26, 27, 43, 45, 53, 54, 45, 53, 54, 86, 90, 91, 107, 109, 90, 91, 107, 109, 173, 181, 182, 214, 218, 181, 182, 214, 218, 346, 347, 363, 365, 429, 437, 363, 365, 429, 437, 693, 694, 726, 730, 858, 859, 875, 726, 730, 858, 859, 875

Offset: 1

T. D. Noe, Jan 12 2011

See A178992 for more details. In binary, the subwords are 0, 1, 01, 10, 11, 010, 011, 101, 110, 0101, 0110, 1010, 1011, 1101,... Converting these numbers to decimal produces this sequence. Except for the initial 0, subwords of the form 0X occur later in the sequence than X. Hence, the second occurrence of a number in this sequence represents the subword having a leading zero. There is a link to a file containing the subwords in binary.

T. D. Noe, Table of n, a(n) for n = 1..1652
T. D. Noe, A list of 1652 subwords in binary

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]

A066744 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +3 for 0, +1 for 1.

Original entry on oeis.org

1, 7, 29, 117, 471, 1885, 7543, 30173, 120693, 482775, 1931101, 7724405, 30897623, 123590493, 494361975, 1977447901, 7909791605, 31639166423, 126556665693, 506226662775, 2024906651101, 8099626604405, 32398506417623

Offset: 1

John McNamara, Jan 16 2002

nonn

Ratio to terms of A028894 tends to 1.23459972586...

Related to A005614, A002450, A037576, A028894.

f[1]={0}; f[2]={1}; f[n_] := f[n]=Join[f[n-1], f[n-2]]; a[1]=1; a[n_] := a[n]=4a[n-1]+3-2f[9][[n]]; a/@Range[1, 30]

A124842 Triangle, row sums = A005614, the rabbit sequence.

Original entry on oeis.org

1, 1, -1, 1, -2, 2, 1, -3, 6, -3, 1, -4, 12, -12, 3, 1, -5, 20, -30, 15, 0, 1, -6, 30, -60, 45, 0, -10, 1, -7, 42, -105, 105, 0, -70, 35, 1, -8, 56, -168, 210, 0, -280, 280, -90

Offset: 1

Gary W. Adamson, Nov 10 2006

			First few rows of the triangle are:
1;
1, -1;
1, -2, 2;
1, -3, 6, -3;
1, -4, 12, -12, 3;
1, -5, 20, -30, 15, 0;
1, -6, 30, -60, 45, 0, -10;
...
4th term of the rabbit sequence (1, 0, 1, 1, 0...) = 1 = sum of row 4 terms: (1, - 3 + 6 - 3).

Cf. A124841.

Binomial transform of the infinite matrix with the diagonalized sequence A124841.

A144023 INVERT transform of the rabbit sequence, A005614.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 17, 29, 49, 82, 138, 234, 394, 663, 1118, 1886, 3179, 5358, 9032, 15227, 25670, 43272, 72945, 122970, 207300, 349456, 589098, 993082, 1674103, 2822138, 4757452, 8019937, 13519716, 22791031, 38420264, 64767451

Offset: 1

Gary W. Adamson, Sep 07 2008

nonn

a(n)/a(n-1) appears to converge to phi.
The sequence generated is first (1, 1, 1, 2, 4, 6, 10,...), but by convention we drop the first "1".
Equals row sums of triangle A144024.

			The operations in steps generate(1, 1, 1, 2...).
a(4) = 4 = (1, 1, 0, 1) dot (1, 1, 1, 2) = (1 + 1 + 0 + 2).

Cf. A144024, A005614.

A144024 Eigentriangle by rows, T(n,k) = A005614(n-k+1)*A144023(k-1).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 4, 1, 0, 1, 2, 0, 6, 0, 1, 0, 2, 4, 0, 10, 1, 0, 1, 0, 4, 6, 0, 17, 1, 1, 0, 20, 6, 10, 0, 29, 0, 1, 1, 0, 4, 0, 10, 17, 0, 4, 9, 1, 0, 1, 2, 0, 6, 0, 17, 29, 0, 82

Offset: 1

Gary W. Adamson, Sep 07 2008

Row sums = A144023, the INVERT transform of the rabbit sequence, A005614.
Left border = A005614.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
  1;
  0, 1;
  1, 0, 1;
  1, 1, 0, 2;
  0, 1, 1, 0, 4;
  1, 0, 1, 2, 0, 6;
  0, 1, 0, 2, 4, 0, 10;
  1, 0, 1, 0, 4, 6, 0, 17;
  1, 1, 0, 2, 0, 6, 10, 0, 29;
  ...;
Row 4 = (1, 1, 0, 2) = termwise product of (1, 1, 0, 1) and (1, 1, 1, 2); where (1, 1, 0, 1) = the first 4 terms of A005614 reversed. (1, 1, 1, 2) = the first 4 terms of shifted A144023.

Cf. A005614, A144023.

Eigentriangle by rows, T(n,k) = A005614(n-k+1)*A144023(k-1).
A005614 = the rabbit sequence, (1, 0, 1, 1, 0, 1, 0, 1,...)
A144023(k-1) = A144023 shifted to (1, 1, 1, 2, 4, 6, 10, 17, 29,...).

cp's OEIS Frontend

A124841 Inverse binomial transform of A005614, the rabbit sequence: (1, 0, 1, 1, 0, ...).

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A089910 Indices n at which blocks (1;1) occur in infinite Fibonacci word, i.e., such that A005614(n-1) = A005614(n-2) = 1.

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A044432 a(n) is the number whose base-2 representation is d(0)d(1)...d(n), where d=A005614 (the infinite Fibonacci word).

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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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A028894 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +1 for 0, +3 for 1.

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A179969 Ordered list in decimal notation of the subwords appearing in the infinite Fibonacci word A005614.

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A066744 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +3 for 0, +1 for 1.

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A124842 Triangle, row sums = A005614, the rabbit sequence.

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