A287869 -id:A287869 - cp's OEIS frontend

A035513 Wythoff array read by falling antidiagonals.

1, 2, 4, 3, 7, 6, 5, 11, 10, 9, 8, 18, 16, 15, 12, 13, 29, 26, 24, 20, 14, 21, 47, 42, 39, 32, 23, 17, 34, 76, 68, 63, 52, 37, 28, 19, 55, 123, 110, 102, 84, 60, 45, 31, 22, 89, 199, 178, 165, 136, 97, 73, 50, 36, 25, 144, 322, 288, 267, 220, 157, 118, 81, 58, 41, 27, 233, 521

Offset: 1

N. J. A. Sloane

T(0,0)=1, T(0,1)=2,...; y^2-x^2-xy

Inverse of sequence A064274 considered as a permutation of the nonnegative integers. - Howard A. Landman, Sep 25 2001

The Wythoff array W consists of all the Wythoff pairs (x(n),y(n)), where x=A000201 and y=A001950, so that W contains every positive integer exactly once. The differences T(i,2j+1)-T(i,2j) form the Wythoff difference array, A080164, which also contains every positive integer exactly once. - Clark Kimberling, Feb 08 2003

For n>2 the determinant of any n X n contiguous subarray of A035513 (as a square array) is 0. - Gerald McGarvey, Sep 18 2004

From Clark Kimberling, Nov 14 2007: (Start)

Except for initial terms in some cases:

(Row 1) = A000045

(Row 2) = A000032

(Row 3) = A006355

(Row 4) = A022086

(Row 5) = A022087

(Row 6) = A000285

(Row 7) = A022095

(Row 8) = A013655 (sum of Fibonacci and Lucas numbers)

(Row 9) = A022112

(Row 10-19) = A022113, A022120, A022121, A022379, A022130, A022382, A022088, A022136, A022137, A022089

(Row 20-28) = A022388, A022096, A022090, A022389, A022097, A022091, A022390, A022098, A022092

(Column 1) = A003622 = AA Wythoff sequence

(Column 2) = A035336 = BA Wythoff sequence

(Column 3) = A035337 = ABA Wythoff sequence

(Column 4) = A035338 = BBA Wythoff sequence

(Column 5) = A035339 = ABBA Wythoff sequence

(Column 6) = A035340 = BBBA Wythoff sequence

Main diagonal = A020941. (End)

The Wythoff array is the dispersion of the sequence given by floor(n*x+x-1), where x=(golden ratio). See A191426 for a discussion of dispersions. - Clark Kimberling, Jun 03 2011

If u and v are finite sets of numbers in a row of the Wythoff array such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. See A160009 (row 1 products), A274286 (row 2), A274287 (row 3), A274288 (row 4). - Clark Kimberling, Jun 17 2016

All columns of the Wythoff array are compound Wythoff sequences. This follows from the main theorem in the 1972 paper by Carlitz, Scoville and Hoggatt. For an explicit expression see Theorem 10 in Kimberling's paper from 2008 in JIS. - Michel Dekking, Aug 31 2017

The Wythoff array can be viewed as an infinite graph over the set of nonnegative integers, built as follows: start with an empty graph; for all n = 0, 1, ..., create an edge between n and the sum of the degrees of all i < n. Finally, remove vertex 0. In the resulting graph, the connected components are chains and correspond to the rows of the Wythoff array. - Luc Rousseau, Sep 28 2017

Suppose that h < k are consecutive terms in a row of the Wythoff array. If k is in an even numbered column, then h = floor(k/tau); otherwise, h = -1 + floor(k/tau). - Clark Kimberling, Mar 05 2020

From Clark Kimberling, May 26 2020: (Start)

For k > = 0, column k shows the numbers m having F(k+1) as least term in the Zeckendorf representation of m. For n >= 1, let r(n,k) be the number of terms in column k that are <= n. Then n/r(n,k) = n/(F(k+1)*tau + F(k)*(n-1)), by Bottomley's formula, so that the limiting ratio is 1/(F(k+1)*tau + F(k)). Summing over all k gives Sum_{k>=0} 1/(F(k+1)*tau + F(k)) = 1. Thus, in the limiting sense:

38.19...% of the numbers m have least term 1;

23.60...% have least term 2;

14.58...% have least term 3;

9.01...% have least term 5, etc. (End)

Named after the Dutch mathematician Willem Abraham Wythoff (1865-1939). - Amiram Eldar, Jun 11 2021

From Clark Kimberling, Jun 04 2025: (Start)

Let u(n) = (T(n,1),T(n,2)) mod 2. The positive integers (A000027) are partitioned into 4 sets (sequences):

{n : u(n) = (0,0)} = (3, 5, 9, 15, 19, 25, 29,...) = 1 + 2*A190429

{n: u(n) = (0,1)} = (2, 6, 12, 16, 18, 22, 28,...) = A191331

{n : u(n) = (1,0)} = (1, 7, 11, 13, 17, 21, 23,...) = A086843

{n: u(n) = (1,1)} = (4, 8, 10, 14, 20, 24, 26,...) = A191330.

Let v(n) = (T(n,1),T(n,2)) mod 3. The positive integers are partitioned into 9 sets (sequences):

{n : v(n) = (0,0)} = (4, 13, 19, 28, 43, 52,...) = 1 + 3*A190434

{n: v(n) = (0,1)} = (3, 12, 27, 36, 42, 51,...) = 3*A140399

{n : v(n) = (0,2)} = (5, 11, 20, 35, 44, 50,...) = 2 + 3*A190439

{n: v(n) = (1,0)} = (9, 18, 24, 33, 48, 57,...) = 3*A140400

{n: v(n) = (1,1)} = (2, 8, 17, 26, 32, 41,...) = A384601

{n : v(n) = (1,2)} = (1, 10, 16, 25, 34, 40,...) = A384602

{n: v(n) = (2,0)} = (14, 23, 29, 38, 47, 53,...) = 2 + 3*A190438

{n: v(n) = (2,1)} = (7, 22, 31, 37, 46, 61,...) = 1 + 3*A190433

{n : v(n) = (2,2)} = (6, 15, 21, 30, 39, 45,...) = 3*A140398.

Conjecture: If m >= 2, then {(T(n,1), T(n,2)) mod m} has cardinality m^2. (End)

			The Wythoff array begins:
   1    2    3    5    8   13   21   34   55   89  144 ...
   4    7   11   18   29   47   76  123  199  322  521 ...
   6   10   16   26   42   68  110  178  288  466  754 ...
   9   15   24   39   63  102  165  267  432  699 1131 ...
  12   20   32   52   84  136  220  356  576  932 1508 ...
  14   23   37   60   97  157  254  411  665 1076 1741 ...
  17   28   45   73  118  191  309  500  809 1309 2118 ...
  19   31   50   81  131  212  343  555  898 1453 2351 ...
  22   36   58   94  152  246  398  644 1042 1686 2728 ...
  25   41   66  107  173  280  453  733 1186 1919 3105 ...
  27   44   71  115  186  301  487  788 1275 2063 3338 ...
  ...
The extended Wythoff array has two extra columns, giving the row number n and A000201(n), separated from the main array by a vertical bar:
0     1  |   1    2    3    5    8   13   21   34   55   89  144   ...
1     3  |   4    7   11   18   29   47   76  123  199  322  521   ...
2     4  |   6   10   16   26   42   68  110  178  288  466  754   ...
3     6  |   9   15   24   39   63  102  165  267  432  699 1131   ...
4     8  |  12   20   32   52   84  136  220  356  576  932 1508   ...
5     9  |  14   23   37   60   97  157  254  411  665 1076 1741   ...
6    11  |  17   28   45   73  118  191  309  500  809 1309 2118   ...
7    12  |  19   31   50   81  131  212  343  555  898 1453 2351   ...
8    14  |  22   36   58   94  152  246  398  644 1042 1686 2728   ...
9    16  |  25   41   66  107  173  280  453  733 1186 1919 3105   ...
10   17  |  27   44   71  115  186  301  487  788 1275 2063 3338   ...
11   19  |  30   49   79   ...
12   21  |  33   54   87   ...
13   22  |  35   57   92   ...
14   24  |  38   62   ...
15   25  |  40   65   ...
16   27  |  43   70   ...
17   29  |  46   75   ...
18   30  |  48   78   ...
19   32  |  51   83   ...
20   33  |  53   86   ...
21   35  |  56   91   ...
22   37  |  59   96   ...
23   38  |  61   99   ...
24   40  |  64   ...
25   42  |  67   ...
26   43  |  69   ...
27   45  |  72   ...
28   46  |  74   ...
29   48  |  77   ...
30   50  |  80   ...
31   51  |  82   ...
32   53  |  85   ...
33   55  |  88   ...
34   56  |  90   ...
35   58  |  93   ...
36   59  |  95   ...
37   61  |  98   ...
38   63  |     ...
   ...
Each row of the extended Wythoff array also satisfies the Fibonacci recurrence, and may be extended to the left using this recurrence backwards.
From _Peter Munn_, Jun 11 2021: (Start)
The Wythoff array appears to have the following relationship to the traditional Fibonacci rabbit breeding story, modified for simplicity to be a story of asexual reproduction.
Give each rabbit a number, 0 for the initial rabbit.
When a new round of rabbits is born, allocate consecutive numbers according to 2 rules (the opposite of many cultural rules for inheritance precedence): (1) newly born child of Rabbit 0 gets the next available number; (2) the descendants of a younger child of any given rabbit precede the descendants of an older child of the same rabbit.
Row n of the Wythoff array lists the children of Rabbit n (so Rabbit 0's children have the Fibonacci numbers: 1, 2, 3, 5, ...). The generation tree below shows rabbits 0 to 20. It is modified so that each round of births appears on a row.
                                                                 0
                                                                 :
                                       ,-------------------------:
                                       :                         :
                       ,---------------:                         1
                       :               :                         :
              ,--------:               2               ,---------:
              :        :               :               :         :
        ,-----:        3         ,-----:         ,-----:         4
        :     :        :         :     :         :     :         :
     ,--:     5     ,--:     ,---:     6     ,---:     7     ,---:
     :  :     :     :  :     :   :     :     :   :     :     :   :
  ,--:  8  ,--:  ,--:  9  ,--:  10  ,--:  ,--:  11  ,--:  ,--:  12
  :  :  :  :  :  :  :  :  :  :   :  :  :  :  :   :  :  :  :  :   :
  : 13  :  : 14  : 15  :  : 16   :  : 17  : 18   :  : 19  : 20   :
The extended array's nontrivial extra column (A000201) gives the number that would have been allocated to the first child of Rabbit n, if Rabbit n (and only Rabbit n) had started breeding one round early.
(End)

John H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.
Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.

Alois P. Heinz, Table of n, a(n) for n = 1..5151
Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21.
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Fibonacci representations, Fib. Quart., Vol. 10, No. 1 (1972), pp. 1-28.
Code Golf Stack Exchange, New Order #5: where Fibonacci and Beatty meet at Wythoff.
J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, and N. J. A. Sloane, On Kimberling Sums and Para-Fibonacci Sequences, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997.
John Conway and Alex Ryba, The extra Fibonacci series and the Empire State Building, Math. Intelligencer, Vol. 38, No. 1 (2016), pp. 41-48.
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18. - _N. J. A. Sloane_, Jun 10 2012
Clark Kimberling, Interspersions.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.
Clark Kimberling, Complementary equations and Wythoff Sequences, JIS, Vol. 11 (2008), Article 08.3.3.
Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123, No. 2 (2016), pp. 267-273.
Stéphane Legendre, Labeled Fibonacci Trees, Fibonacci Quart. 53 (2015), no. 2, 152-167.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See pages 1-2.
Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, Vol. 41 (2013), pp. 175-192.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences
Sam Vandervelde, On the divisibility of Fibonacci sequences by primes of index two, The Fibonacci Quarterly, Vol. 50, No. 3 (2012), pp. 207-216. See Figure 1.
Eric Weisstein's World of Mathematics, Wythoff Array.
Wikipedia, Wythoff array.
Pedro Zanetti, C++ code snippet, for generating this sequence.
Index entries for sequences that are permutations of the natural numbers

See comments above for more cross-references.
Cf. A003622, A064274 (inverse), A083412 (transpose), A000201, A001950, A080164, A003603, A265650, A019586 (row that contains n).
For two versions of the extended Wythoff array, see A287869, A287870.

W:= proc(n,k) Digits:= 100; (Matrix([n, floor((1+sqrt(5))/2* (n+1))]). Matrix([[0,1], [1,1]])^(k+1))[1,2] end: seq(seq(W(n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Aug 18 2008
A035513 := proc(r, c)
    option remember;
    if c = 1 then
        A003622(r) ;
    else
        A022342(1+procname(r, c-1)) ;
    end if;
end proc:
seq(seq(A035513(r,d-r),r=1..d-1),d=2..15) ; # R. J. Mathar, Jan 25 2015

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; Table[ W[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten

T(n,k)=(n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)
for(k=0,9,for(n=1,k, print1(T(n,k+1-n)", "))) \\ Charles R Greathouse IV, Mar 09 2016

from sympy import fibonacci as F, sqrt
import math
tau = (sqrt(5) + 1)/2
def T(n, k): return F(k + 1)*int(math.floor(n*tau)) + F(k)*(n - 1)
for n in range(1, 11): print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 23 2017

from math import isqrt, comb
from gmpy2 import fib2
def A035513(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    b, c = fib2(a-x+2)
    return b*(x+isqrt(5*x*x)>>1)+c*(x-1) # Chai Wah Wu, Jun 26 2025

T(n, k) = Fib(k+1)*floor[n*tau]+Fib(k)*(n-1) where tau = (sqrt(5)+1)/2 = A001622 and Fib(n) = A000045(n). - Henry Bottomley, Dec 10 2001

T(n,-1) = n-1. T(n,0) = floor(n*tau). T(n,k) = T(n,k-1) + T(n,k-2) for k>=1. - R. J. Mathar, Sep 03 2016

Comments about the extended Wythoff array added by N. J. A. Sloane, Mar 07 2016

A287870 The extended Wythoff array (the Wythoff array with two extra columns) read by antidiagonals downwards.

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 2, 4, 4, 3, 3, 7, 6, 6, 4, 5, 11, 10, 9, 8, 5, 8, 18, 16, 15, 12, 9, 6, 13, 29, 26, 24, 20, 14, 11, 7, 21, 47, 42, 39, 32, 23, 17, 12, 8, 34, 76, 68, 63, 52, 37, 28, 19, 14, 9, 55, 123, 110, 102, 84, 60, 45, 31, 22, 16, 10, 89, 199, 178, 165, 136, 97, 73, 50, 36, 25, 17, 11

Offset: 1

N. J. A. Sloane, Jun 14 2017

From Peter Munn, Apr 28 2025: (Start)
Each row in the Wythoff array, A035513, and this extended array satisfies the Fibonacci recurrence; that is each term after the first 2 is the sum of the preceding 2 terms.
We use F_i to denote the i-th Fibonacci term, A000045(i). In particular, we refer below to F_0 = 0, F_1 = 1 and F_2 = 1 several times. Note that to fully understand the description of the relationship between neighboring columns it is important to distinguish F_1 and F_2, although they have the same integer value. Similarly, the identity of an array term should be understood here as including its position in the array, not only its integer value.
The terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. With this map each term is the sum of its subset image. See the table in the examples.
Full description of the mapping with its relationship to A035513:
The (unextended) Wythoff array A035513 includes every positive integer exactly once. So, using the Zeckendorf representation (see link below), the array terms map 1:1 to nonempty finite subsets of the Fibonacci terms from F_2 onwards -- more precisely, onto those that do not include both F_i and F_{i+1} for any i. (Again, each array term is the sum of the Fibonacci numbers from the relevant subset.)
As shown in the Kimberling 1995 link, when we proceed from one term to the next in a row, the indices of the Fibonacci terms in the corresponding subset are incremented. When we proceed leftwards, the indices are decremented, with the subsets for the leftmost column being those that include F_2.
And when we add 2 columns on the left of the Wythoff array, the mapping continues to decrement the indices, so the corresponding extra subsets have F_0 (new leftmost column) or F_1 as their first Fibonacci term.
Thus the terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. The leftmost column is the nonnegative integers: if we were to remove F_0 (value 0) from the subset for an integer in this column, the subset would form the Zeckendorf representation of the integer, as subsets do in the unextended array.
(End)

			The extended Wythoff array is the Wythoff array with two extra columns, giving the row number n and A000201(n), separated from the main array by a vertical bar:
   0   1 |  1   2   3    5    8   13   21   34    55    89   144 ...
   1   3 |  4   7  11   18   29   47   76  123   199   322   521 ...
   2   4 |  6  10  16   26   42   68  110  178   288   466   754 ...
   3   6 |  9  15  24   39   63  102  165  267   432   699  1131 ...
   4   8 | 12  20  32   52   84  136  220  356   576   932  1508 ...
   5   9 | 14  23  37   60   97  157  254  411   665  1076  1741 ...
   6  11 | 17  28  45   73  118  191  309  500   809  1309  2118 ...
   7  12 | 19  31  50   81  131  212  343  555   898  1453  2351 ...
   8  14 | 22  36  58   94  152  246  398  644  1042  1686  2728 ...
   9  16 | 25  41  66  107  173  280  453  733  1186  1919  3105 ...
  10  17 | 27  44  71  115  186  301  487  788  1275  2063  3338 ...
  11  19 | 30  49  79 ...
  12  21 | 33  54  87 ...
  13  22 | 35  57  92 ...
  14  24 | 38  62 ...
  15  25 | 40  65 ...
  16  27 | 43  70 ...
  17  29 | 46  75 ...
  18  30 | 48  78 ...
  19  32 | 51  83 ...
  20  33 | 53  86 ...
  21  35 | 56  91 ...
  22  37 | 59  96 ...
  23  38 | 61  99 ...
  24  40 | 64 ...
  25  42 | 67 ...
  26  43 | 69 ...
  27  45 | 72 ...
  28  46 | 74 ...
  29  48 | 77 ...
  30  50 | 80 ...
  31  51 | 82 ...
  32  53 | 85 ...
  33  55 | 88 ...
  34  56 | 90 ...
  35  58 | 93 ...
  36  59 | 95 ...
  37  61 | 98 ...
  38  63 | ...
  ...
From _Peter Munn_, Sep 12 2022: (Start)
In the table below, the array terms are shown in the small box at the bottom right of the cells. At the top of each cell is shown a pattern of Fibonacci terms, with "*" indicating a Fibonacci term that appears below it. Those Fibonacci terms sum to the array term. The pattern never includes "**", which would indicate 2 consecutive Fibonacci terms. Note that a Fibonacci term shown as "1" in the 2nd column is F_1, so it may accompany "2", which is F_3. In other columns a Fibonacci term shown as "1" is F_2 and may not accompany "2".
+----------+-----------+------------+------------+------------+
|      *   |      *    |       *    |       *    |       *    |
|      0 __|      1 ___|       1 ___|       2 ___|       3 ___|
|       |0 |       | 1 |        | 1 |        | 2 |        | 3 |
|----------+-----------+------------+------------+------------|
|    * *   |    * *    |     * *    |     * *    |     * *    |
|      0 __|      1 ___|       1 ___|       2 ___|       3 ___|
|    1  |1 |    2  | 3 |     3  | 4 |     5  | 7 |     8  |11 |
|----------+-----------+------------+------------+------------|
|   *  *   |   *  *    |    *  *    |    *  *    |    *  *    |
|   2  0 __|   3  1 ___|    5  1 ___|    8  2 ___|   13  3 ___|
|       |2 |       | 4 |        | 6 |        |10 |        |16 |
|----------+-----------+------------+------------+------------|
|  *   *   |  *   *    |   *   *    |   *   *    |   *   *    |
|      0 __|      1 ___|       1 ___|       2 ___|       3 ___|
|  3    |3 |  5    | 6 |   8    | 9 |  13    |15 |  21    |24 |
|----------+-----------+------------+------------+------------|
|  * * *   |  * * *    |   * * *    |   * * *    |   * * *    |
|      0   |      1    |       1    |       2    |       3    |
|    1   __|    2   ___|     3   ___|     5   ___|     8   ___|
|  3    |4 |  5    | 8 |   8    |12 |  13    |20 |  21    |32 |
|----------+-----------+------------+------------+------------|
| *    *   | *    *    |  *    *    |  *    *    |  *    *    |
|      0 __|      1 ___|       1 ___|       2 ___|       3 ___|
| 5     |5 | 8     | 9 | 13     |14 | 21     |23 | 34     |37 |
|----------+-----------+------------+------------+------------|
| *  * *   | *  * *    |  *  * *    |  *  * *    |  *  * *    |
|      0 __|      1 ___|       1 ___|       2 ___|       3 ___|
| 5  1  |6 | 8  2  |11 | 13  3  |17 | 21  5  |28 | 34  8  |45 |
|----------+-----------+------------+------------+------------|
| * *  *   | * *  *    |  * *  *    |  * *  *    |  * *  *    |
|   2  0 __|   3  1 ___|    5  1 ___|    8  2 ___|   13  3 ___|
| 5     |7 | 8     |12 | 13     |19 | 21     |31 | 34     |50 |
+----------+-----------+------------+------------+------------+
If we replace the Fibonacci terms 0, 1, 1, 2, 3, 5, ... in the main part of the cells with the powers of 2 (1, 2, 4, ...) the sums in the small boxes become the terms of A356875. From this may be seen a relationship to A054582.
- - - - -
Each row of the extended Wythoff array satisfies the Fibonacci recurrence, and may be further extended to the left using this recurrence backwards:
... -1   1   0   1 |  1   2    3    5 ...
... -1   2   1   3 |  4   7   11   18 ...
...  0   2   2   4 |  6  10   16   26 ...
...  0   3   3   6 |  9  15   24   39 ...
...  0   4   4   8 | 12  20   32   52 ...
...  1   4   5   9 | 14  23   37   60 ...
...  1   5   6  11 | 17  28   45   73 ...
...  2   5   7  12 | 19  31   50   81 ...
...  2   6   8  14 | 22  36   58   94 ...
    ...
...  5  10  15  25 | 40  65  105  170 ...
    ...
Note that multiples (*2, *3 and *4) of the top (Fibonacci sequence) row appear a little below, but shifted 2 columns to the left. Larger multiples appear further down and shifted further to the left, starting with row 15, where the terms are 5 times those in the top row and shifted 4 columns leftwards.
(End)

Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21.
John Conway and Alex Ryba, The extra Fibonacci series and the Empire State Building, Math. Intelligencer 38 (2016), no. 1, 41-48. See preview, at ResearchGate.
Encyclopedia of Mathematics, Zeckendorf representation
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.

Subtables: A035513 (the Wythoff array), A287869.
Related as a subtable of A357316 as A054582 is to A130128 (as a square).
Cf. A000045, A000201.
See A014417 for sequences related to Zeckendorf representation.
See the formula section for the relationships with A003622, A022341, A054582, A356874, A356875.

From Peter Munn, Apr 29 2025: (Start)
A(n,k) = A356874(floor(m/2)), where m = A356875(n-1, k-1) = A054582(k-1, (A022341(n-1)-1)/2).
A(n,k) = A357316(A003622(n), k-1).
(End)

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)).

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cp's OEIS Frontend

A035513 Wythoff array read by falling antidiagonals.

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A287870 The extended Wythoff array (the Wythoff array with two extra columns) read by antidiagonals downwards.

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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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