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

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

Author

N. J. A. Sloane, Jun 14 2017

Keywords

• nonn
• tabl

Comments

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)

Examples

			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)
		

Links

• 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.

Crossrefs

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.

Formula

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)