A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

0, 0, 1, 0, 2, 1, 0, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Author

Rémy Sigrist, Apr 02 2023

Keywords

• nonn
• base
• tabf

Comments

We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.

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

This sequence can also be seen as an irregular table T(n, k), n > 0, k = 1..A000045(n), where T(n, k) = A000045(n) - k.

a(n-1) for n>=1 is the starting position of the first occurrence of one of the longest words w in the Fibonacci word A003849 such that no length-n factor of w is repeated. The length of such words is 2n. (See links) - Gandhar Joshi, Mar 19 2024

Examples

			For n = 42:
- using F(k) = A000045(k),
- the dual Zeckendorf representation of 42 is F(8) + F(7) + F(5) + F(3) + F(2),
- the numbers F(6) and F(4) are missing,
- so a(42) = F(6) + F(4) = 8 + 3 = 11.
.
As an irregular triangle the sequence begins:
     0;
     0;
     1,  0;
     2,  1,  0;
     4,  3,  2, 1, 0;
     7,  6,  5, 4, 3, 2, 1, 0;
    12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0;
    ...
		

Links

• Gandhar Joshi, Walnut code and details
• Index entries for sequences related to Zeckendorf expansion of n

Crossrefs

Cf. A000045, A003754, A022290, A035327, A072649, A130312, A132665, A194029, A356771.

Programs

PARI

for (n = 1, 9, for (k = 1, f = fibonacci(n), print1 (f-k", ")))

Formula

a(n) = A000045(A072649(n)) - A194029(n) for n > 0.

a(n) = A130312(n) - A194029(n) for n > 0.