A134859 -id:A134859 - cp's OEIS frontend

A246354 Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(1)..s(n) occurs for the k-th time.

1, 3, 1, 4, 4, 1, 6, 6, 4, 1, 8, 9, 6, 6, 1, 9, 12, 9, 9, 6, 1, 11, 14, 12, 14, 9, 6, 1, 12, 17, 14, 19, 14, 9, 9, 1, 14, 19, 17, 22, 19, 14, 14, 9, 1, 16, 22, 19, 27, 22, 19, 22, 14, 9, 1, 17, 25, 22, 30, 27, 22, 30, 22, 14, 9, 1, 19, 27, 25, 35, 30, 27, 35

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 lower Wythoff sequence, A000201 gives the positions of 0 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(1)..s(1) is simply 0, which occurs at positions 1,3,4,6,8,... as in row 1 of T.  For n = 5, the block s(1)..s(5) is 0,1,0,0,1, which occurs at positions 1,6,9,14,19, ...
The first 7 rows follow:
1 .. 3 .. 4 ... 6 ... 8 ... 9 ... 11 .. 12 ...
1 .. 4 .. 6 ... 9 ... 12 .. 14 .. 17 .. 19 ...
1 .. 4 .. 6 ... 9 ... 12 .. 14 .. 17 .. 19 ...
1 .. 6 .. 9 ... 14 .. 19 .. 22 .. 27 .. 30 ...
1 .. 6 .. 9 ... 14 .. 19 .. 22 .. 27 .. 30 ...
1 .. 6 .. 9 ... 14 .. 19 .. 22 .. 27 .. 30 ...
1 .. 9 .. 14 .. 22 .. 30 .. 35 .. 43 .. 48 ...

Cf. A003849, A246355.

z = 1000; s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, 12]]; Flatten[Position[s, 0]];  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[1, 1 + k] &], z2]; Column[Table[t[k], {k, 0, z2}]] (* A246354, array *)
w[n_, k_] := t[n][[k + 1]]; Table[w[n - k, k], {n, 0, z2 - 1}, {k, n, 0, -1}] // Flatten (*  A246354, sequence *)

First row: A000201 (lower Wythoff numbers);
next 2 rows:  A003622 (Wythoff AA numbers);
next 3 rows:  A134859 (Wythoff AAA numbers);
next 5 rows:  A151915 (Wythoff AAAA 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.

A340244 Wythoff-B array read by antidiagonals.

Original entry on oeis.org

4, 7, 12, 11, 20, 17, 18, 32, 28, 25, 29, 52, 45, 41, 33, 47, 84, 73, 66, 54, 38, 76, 136, 118, 107, 87, 62, 46, 123, 220, 191, 173, 141, 100, 75, 51, 199, 356, 309, 280, 228, 162, 121, 83, 59, 322, 576, 500, 453, 369, 262, 196, 134, 96, 67, 521, 932, 809

Offset: 1

Clark Kimberling, Jan 02 2021

The Wythoff array, A134859, consists of columns AA, BA, ABA, BBA, ABBA, BBBA, ... The Wythoff-B array consists of columns AAB, BAB, ABAB, BBAB, ABBAB, BBBAB, ... , formed by suffixing B to the column designations for A134859. Column k shows the numbers whose Zeckendorf representation has least terms F(k+1) and F(k+2), where F = A000045, the Fibonacci numbers. The rows are interspersed, in the sense that the order array (A340245) of the Wythoff-B array is an interspersion.

			Corner:
   4    7    11    18    29    47   76     123    199
  12   20    32    52    84   136   220    356    576
  17   28    45    73   118   191   309    500    809
  25   41    66   107   173   280   453    733   1186
  33   54    87   141   228   369   597    966   1563
  38   62   100   162   262   424   686   1110   1796
  46   75   121   196   317   513   830   1343   2173
  51   83   134   217   351   568   919   1487   2406

Cf. A000045, A000201, A001950, A134859, A340245.

r = GoldenRatio; f[n_] := Fibonacci[n];
a[n_] := Floor[r*n]; b[n_] := Floor[r^2*n];
c[n_] := a[a[b[n]]]; d[n_] := b[a[b[n]]];
w[n_, k_] := f[k - 2] c[n] + f[k - 1] d[n];
Grid[Table[w[n, k], {n, 1, 15}, {k, 1, 15}]] (* A340244 array *)
Table[w[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten  (* A340244 sequence *)

For n >=1 and k >= 1, w(n,k) = F(k-2)*A(A(B(n))) + F(k-1)*B(A(B(n))), where A(n) = floor(n*phi), B(n) = floor(n*phi^2); i.e., A = A000201, B= A001950, these being the lower and upper Wythoff sequences. (Note that F(-1) = 1, F(0) = 0.)

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).

cp's OEIS Frontend

A246354 Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(1)..s(n) 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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A340244 Wythoff-B array read by antidiagonals.

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

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