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}.

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

0,12

Comments

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).

Links

Table of n, a(n) for n=0..77.

Encyclopedia of Mathematics, Zeckendorf representation

Formula

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)).

Example

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).

Prog

(PARI)
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

Crossrefs

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.

See the formula section for the relationship to A003714, A022342, A135090, A356874.

Sequence in context: A076451 A230536 A306257 * A108839 A114898 A223903

Adjacent sequences: A357313 A357314 A357315 * A357317 A357318 A357319

Keyword

nonn,tabl

Author

Peter Munn, Sep 23 2022

Status

approved