A361756 Irregular triangle T(n, k), n >= 0, k = 1..A361757(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the dual Zeckendorf representation of k also appear in that of n.

0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 4, 0, 2, 5, 0, 1, 2, 3, 4, 5, 6, 0, 2, 7, 0, 1, 2, 3, 7, 8, 0, 1, 4, 9, 0, 2, 5, 7, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 4, 12, 0, 2, 5, 13, 0, 1, 2, 3, 4, 5, 6, 12, 13, 14, 0, 2, 7, 15, 0, 1, 2, 3, 7, 8, 15, 16

Offset

0,5

Comments

In other words, the n-th row lists the numbers k such that A003754(1+n) AND A003754(1+k) = A003754(1+k) (where AND denotes the bitwise AND operator).

The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..9956 (rows for n = 0..377 flattened)

Rémy Sigrist, PARI program

Index entries for sequences related to Zeckendorf expansion of n

Formula

T(n, 1) = 0.

T(n, 2) = A003842(n - 1) for any n > 0.

T(n, A361757(n)) = n.

Example

Triangle T(n, k) begins:
  n   n-th row
  --  -------------------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 1, 2, 3
   4  0, 1, 4
   5  0, 2, 5
   6  0, 1, 2, 3, 4, 5, 6
   7  0, 2, 7
   8  0, 1, 2, 3, 7, 8
   9  0, 1, 4, 9
  10  0, 2, 5, 7, 10
  11  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
  12  0, 1, 4, 12

Prog

(PARI) See Links section.

Crossrefs

See A361755 for a similar sequence.

Cf. A003754, A003842, A356771, A361757.

Sequence in context: A233292 A108456 A089107 * A364912 A321449 A180279

Adjacent sequences: A361753 A361754 A361755 * A361757 A361758 A361759

Keyword

nonn,base,tabf

Author

Rémy Sigrist, Mar 23 2023

Status

approved