A151915 -id:A151915 - 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.)

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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