A035336 -id:A035336 - cp's OEIS frontend

A187951 Positions of 0 in A187950; complement of A187952.

2, 4, 5, 7, 10, 12, 13, 15, 17, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 38, 39, 41, 44, 46, 47, 49, 51, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 93, 94, 96, 99, 101, 102, 104, 106, 107, 109, 112, 114, 115, 117, 120, 122, 123, 125, 127, 128, 130, 133, 135, 136, 138, 140, 141, 143, 146, 148, 149, 151, 154

Offset: 1

Clark Kimberling, Mar 16 2011

nonn

See A187950.

Union of the Wythoff AAB-numbers A134860, the Wythoff BA-numbers A035336 and the Wythoff BB-numbers A101864. - Michel Dekking, Sep 14 2016

Cf. A187950, A187952.

Mathematica
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(See A187950.)
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A246355 Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(2)..s(n+1) occurs for the k-th time.

Original entry on oeis.org

2, 5, 2, 7, 5, 2, 10, 7, 7, 2, 13, 10, 10, 7, 2, 15, 13, 15, 10, 7, 2, 18, 15, 20, 15, 10, 10, 2, 20, 18, 23, 20, 15, 15, 10, 2, 23, 20, 28, 23, 20, 23, 15, 10, 2, 26, 23, 31, 28, 23, 31, 23, 15, 10, 2, 28, 26, 36, 31, 28, 36, 31, 23, 15, 10, 2, 31, 28, 41

Offset: 1

Clark Kimberling, Aug 24 2014

Assuming that every row of T is infinite, each row contains the next row as a proper subsequence. Row 1 of A246354 and row 1 of A246355 partition the positive integers.

			The upper Wythoff sequence, A001950 gives the positions of 1 in A003849, which begins thus:  0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1.  For n = 1, the block s(2)..s(2) is simply 1, which occurs at positions 2,5,7,10,13,... as in row 1 of T.  For n = 5, the block s(2)..s(6) is 1,0,0,1,0 which occurs at positions 2,7,10,15,20,23, ...
The first 6 rows follow:
2 .. 5 .. 7 ... 10 .. 13 .. 15 .. 18 ...
2 .. 5 .. 7 ... 10 .. 13 .. 15 .. 18 ...
2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
2 .. 10 . 15 .. 23 .. 31 .. 36 .. 44 ...

Cf. A003849, A246354.

z = 1000; s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, 12]]; Flatten[Position[s, 1]]; b[m_, n_] := b[m, n] = Take[s, {m, n}]; z1 = 500; z2 = 12; t[k_] := t[k] = Take[Select[Range[1, z1], b[#, # + k] == b[2, 2 + k] &], z2]; Column[Table[t[k], {k, 0, z2}]](* A246355, array *)
w[n_, k_] := t[n][[k + 1]]; Table[w[n - k, k], {n, 0, z2 - 1}, {k, n, 0, -1}] // Flatten (* A246355, sequence *)

First 2 rows:  A001950 (upper Wythoff numbers);
next 3 rows:  A035336 (Wythoff BA numbers);
next 5 rows:  A134861 (Wythoff BAA numbers);
next 8 rows:  (Wythoff BAAA numbers).
(The patterns continue; in particular the number of identical consecutive rows is always a Fibonacci number, as in A000045.)

A276757 Infinite Fibonacci word on the alphabet {1,2,3,4,5}.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2

Offset: 1

Michel Dekking, Sep 17 2016

nonn

Start with the infinite Fibonacci word A003849, which is 0100101001001010010... and replace each 0 by 1,2,3 and each 1 by 4,5.

The unique fixed point of the 4-block Fibonacci substitution 1 -> 12, 2 -> 3, 3 -> 45, 4 -> 12, 5 -> 3. Here the 4-blocks are coded as 0100 <-> 1, 1001 <-> 2, 0010 <-> 3, 0101 <-> 4, 1010 <-> 5.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, arXiv preprint arXiv:1608.04487 [math.DS], 2016.

Cf. A000201, A001950, A003849, A134859, A035336, A003623, A134860, A101864, A270788, A138967.

Let A(n) = floor(n*phi), B(n) = n + floor(n*phi), i.e., A and B are the lower and upper Wythoff sequences, A = A000201, B = A001950. Then a(n) = 1 if n = A(A(A(k))) for some k; a(n) = 2 if n = B(A(k)) for some k; a(n) = 3 if n = A(B(k)) for some k; a(n) = 4 if n = A(A(B(k))) for some k; a(n) = 5 if n = B(B(k)) for some k.

A080746 Inverse Aronson transform of lower Wythoff sequence A000201.

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 57, 59, 61, 62, 64, 65, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 90, 91, 93, 95, 96, 98, 99

Offset: 1

Matthew Vandermast, Mar 08 2003

nonn

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Union of A003622 and A035336, omitting 2. Cf. A000201, A080760.

Same as A026273 except for initial terms.

Consists of numbers [phi[phi*k]] (k >= 1) and [phi[phi^2*k]]-1 (k >= 2), where phi = (1+sqrt(5))/2. - N. J. A. Sloane, Mar 10 2003

A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 2, 1, 0, 3, 3, 3, 3, 2, 0, 5, 5, 5, 4, 3, 2, 0, 8, 8, 8, 7, 5, 4, 3, 0, 13, 13, 13, 11, 8, 6, 4, 3, 0, 21, 21, 21, 18, 13, 10, 7, 5, 3, 0, 34, 34, 34, 29, 21, 16, 11, 8, 6, 4, 0, 55, 55, 55, 47, 34, 26, 18, 13, 9, 6, 4

Offset: 0

Peter Munn, Sep 23 2022

Note the Zeckendorf representation of 0 is taken to be the empty sum.
The Wythoff array A035513 is the subtable formed by rows 3, 11, 16, 24, 32, ... (A035337). If, instead, we use rows 2, 7, 10, 15, 20, ... (A035336) or 1, 4, 6, 9, 12, ... (A003622), we get the Wythoff array extended by 1 column (A287869) or 2 columns (A287870) respectively.
Similarly, using A035338 truncates by 1 column; and in general if S_k is column k of the Wythoff array then the rows here numbered by S_k form an array A_k that starts with column k-2 of the Wythoff array. (A_0 and A_1 are the 2 extended arrays mentioned above.) As every positive integer occurs exactly once in the Wythoff array, every row except row 0 of A(.,.) is a row of exactly one such A_k.
Columns 4 onwards match certain columns of the multiplication table for Knuth's Fibonacci (or circle) product (extended variant - see A135090 and formula below).
For k > 0, the first row to contain k is A348853(k).

			Example for n = 4, k = 3. The Zeckendorf representation of 4 is F_4 + F_2 = 3 + 1. So the values of i in the sums in the definition are 4 and 2; hence A(4,3) = Sum_{i = 2,4} F_{i+k-2} = F_{4+3-2} + F_{2+3-2} = F_5 + F_3 = 5 + 2 = 7.
Square array A(n,k) begins:
   n\k| 0   1    2    3    4    5    6
  ----+--------------------------------
   0  | 0   0    0    0    0    0    0  ...
   1* | 0   1    1    2    3    5    8  ...
   2  | 1   1    2    3    5    8   13  ...
   3  | 1   2    3    5    8   13   21  ...
   4* | 1   3    4    7   11   18   29  ...
   5  | 2   3    5    8   13   21   34  ...
   6* | 2   4    6   10   16   26   42  ...
   7  | 3   4    7   11   18   29   47  ...
   8  | 3   5    8   13   21   34   55  ...
   9* | 3   6    9   15   24   39   63  ...
  10  | 4   6   10   16   26   42   68  ...
  11  | 4   7   11   18   29   47   76  ...
  12* | 4   8   12   20   32   52   84  ...
  ...
The asterisked rows form the start of the extended Wythoff array (A287870).

Encyclopedia of Mathematics, Zeckendorf representation

Columns, some differing initially: A005206 (1), A022342 (3), A026274 (4), A101345 (5), A101642 (6).
Rows: A000045 (1), A000204 (4).
Related to subtable A287870 as A130128 (as a square) is to A054582.
Other subtables: A035513, A287869.
See the comments for the relationship to A003622, A035336, A035337, A035338, A348853.
See the formula section for the relationship to A003714, A022342, A135090, A356874.

A5206(m) = if(m>0,m-A5206(A5206(m-1)),0)
A(n,k) = if(k==2,n, if(k==1,A5206(n), if(k==0,n-A5206(n), A(n,k-2)+A(n,k-1)))) \\ simple encoding of formulas, not efficient

For n >= 0, k >= 0 unless stated otherwise:
A(n,k) = A356874(floor(A003714(n)*2^(k-1))).
A(n,1) = A005206(n).
A(n,2) = n.
A(n,k+2) = A(n,k) + A(n,k+1).
A(A022342(n+1),k) = A(n,k+1).
For k >= 4, A(n,k) = A135090(n,A000045(k-2)).

A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

Original entry on oeis.org

6, 19, 27, 40, 53, 61, 74, 82, 95, 108, 116, 129, 142, 150, 163, 171, 184, 197, 205, 218, 226, 239, 252, 260, 273, 286, 294, 307, 315, 328, 341, 349, 362, 375, 383, 396, 404, 417, 430, 438, 451, 459, 472, 485, 493, 506, 519, 527, 540, 548, 561, 574, 582, 595, 603

Offset: 1

A.H.M. Smeets, Apr 25 2024

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A000045, A005614, A014417.
Tree of Zeckendorf subsequences of positive integers partitioned by their suffix part S (except initial term or offset in some cases). $ is the empty string. length(S) =
0        1        2        3        4        5        6       7
----------------------------------------------------------------------
$:       0:       00:      000:     0000:    00000:   000000:
A000027  A022342  A026274  A101345  A101642  notOEIS  notOEIS
                                                      100000: 0100000:
                                                      A035340  <------
                                             10000:
                                             A035339
                                    1000:    01000:
                                    A035338  <------
                  10:      010:     0010:
                  A035336  <------  A134861
                                    1010:    01010:
                                    A134863  <------
                           100:     0100:
                           A035337  <------
         1:       01:      001:     0001:
         A003622  <------  A134859  A151915
                                    1001:    01001:
                                    A372302  <------
                           101:     0101:
                           A134860  <------
Suffixes 10^n, where ^ means n times repeated concatenation, are the (n+1)-th columns in the Wythoff array A083412 and A035513 (n >= 0).

Equals {A134859}\{A151915}.
a(n) = A134863(n) - 1 = A035338(n) + 1.
a(n) = a(n-1) + 8 + 5*A005614(n-2) = a(n-1) + F(6) + F(5)*A005614(n-2), n > 1, where F(k) is the k-th Fibonacci number (A000045).

A383999 Sequence obtained by replacing 3-term subwords of A003849 by 0,1,2,3 as described in Comments.

Original entry on oeis.org

1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2

Offset: 1

Clark Kimberling, May 23 2025

nonn

The four 3-term subwords of A003849 are 0,0,1; 0,1,0; 1,0,0; 1,0,1. These are coded as 0,1,2,3, respectively, and then these numbers replace the corresponding subwords in A003849. In the resulting sequence, the positions of 0,1,2,3 are given by A003623, A003622, A035336, A101864, which are the x-Wythoff sequences for x = AB, AA, BA, BB, respectively. The corresponding limiting densities are sqrt(5)-2, (3-sqrt(5))/2, sqrt(5)-2, (7-3*sqrt(5))/2.

			Starting with A003849 = (0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,...), the successive 3-term subwords are 0,1,0; 1,0,0; 0,0,1; 0,1,0; 1,0,1; ..., which code as 1,2,0,1,3,... .

Cf. A003849, A003623, A003622, A035336, A101864, A035513, A381848.

Partition[Nest[Flatten[#/. {0->{0,1},1->{0}}]&,{0},10],3,1]/.Thread[{{0,0,1},{0,1,0},{1,0,0},{1,0,1}}->{0,1,2,3}]  (* Peter J. C. Moses, May 22 2025 *)

cp's OEIS Frontend

A187951 Positions of 0 in A187950; complement of A187952.

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A246355 Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(2)..s(n+1) occurs for the k-th time.

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A276757 Infinite Fibonacci word on the alphabet {1,2,3,4,5}.

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A080746 Inverse Aronson transform of lower Wythoff sequence A000201.

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A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.

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A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

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A383999 Sequence obtained by replacing 3-term subwords of A003849 by 0,1,2,3 as described in Comments.

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