A361789 A(n, k) is the sum of the distinct terms in the dual Zeckendorf representations of n or of k; square array A(n, k) read by antidiagonals, n, k >= 0.

0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 6, 6, 3, 6, 6, 6, 7, 6, 5, 6, 6, 5, 6, 7, 8, 8, 6, 6, 4, 6, 6, 8, 8, 9, 8, 7, 6, 6, 6, 6, 7, 8, 9, 10, 9, 8, 8, 6, 5, 6, 8, 8, 9, 10, 11, 11, 11, 8, 11, 6, 6, 11, 8, 11, 11, 11, 12, 11, 10, 11, 11, 10, 6, 10, 11, 11, 10, 11, 12

Offset

0,4

Comments

The dual Zeckendorf representation corresponds to the lazy Fibonacci representation (see A356771 for further details).

Links

Table of n, a(n) for n=0..90.

Rémy Sigrist, Colored representation of the table for x, y < Fibonacci(16) - 1 (where the color is function of A(x, y))

Rémy Sigrist, PARI program

Index entries for sequences related to Zeckendorf expansion of n

Formula

A(n, k) = A022290(A003754(n+1) OR A003754(k+1)) (where OR denotes the bitwise OR operator, A004198).

A(n, k) = A(k, n).

A(n, 0) = n.

A(n, n) = n.

A(A(m, n), k) = A(m, A(n, k)).

A(A(n, k), n) = A(n, k).

A(n, A361756(n, k)) = n.

Example

Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
  ----+-------------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
    1 |  1   1   3   3   4   6   6   8   8   9  11  11  12  14
    2 |  2   3   2   3   6   5   6   7   8  11  10  11  14  13
    3 |  3   3   3   3   6   6   6   8   8  11  11  11  14  14
    4 |  4   4   6   6   4   6   6  11  11   9  11  11  12  14
    5 |  5   6   5   6   6   5   6  10  11  11  10  11  14  13
    6 |  6   6   6   6   6   6   6  11  11  11  11  11  14  14
    7 |  7   8   7   8  11  10  11   7   8  11  10  11  19  18
    8 |  8   8   8   8  11  11  11   8   8  11  11  11  19  19
    9 |  9   9  11  11   9  11  11  11  11   9  11  11  17  19
   10 | 10  11  10  11  11  10  11  10  11  11  10  11  19  18
   11 | 11  11  11  11  11  11  11  11  11  11  11  11  19  19
   12 | 12  12  14  14  12  14  14  19  19  17  19  19  12  14
   13 | 13  14  13  14  14  13  14  18  19  19  18  19  14  13

Prog

(PARI) See Links section.

Crossrefs

Cf. A003754, A003986, A022290, A334348, A356771, A356969, A361756.

Sequence in context: A123243 A037193 A291615 * A343040 A343044 A003986

Adjacent sequences: A361786 A361787 A361788 * A361790 A361791 A361792

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Mar 24 2023

Status

approved