A354321 Digit above the least significant 01 digit pair in the Zeckendorf representation of n.

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1

Offset

1

Comments

The Zeckendorf representation of n, as digits, is A189920 or A014417 and when n = "... d 01 00..00", a(n) = d.

If n is a Fibonacci number (A000045) then n = 1 00..00 and initial 0 digits are understood so that a(n) = 0.

Fibbinary (A003714) encodes Zeckendorf digits in bits and there a(n) is the bit two places above the least significant 1-bit, so a(n) = A086483(A003714(n)).

This sequence is morphic since it can be formed from the fixed point of a morphism (of 7 symbols) by a symbol-to-symbol mapping.

The formulas below using mod phi work since the possible least significant 3 digits fall into disjoint ranges mod phi (since higher terms are Fibonacci(k) == (-1/phi)^(k-1) (mod phi)).

Links

Kevin Ryde, Table of n, a(n) for n = 1..10000

Formula

a(n) = 1 if r < 2*phi-3, or a(n) = 0 if 2*phi-3 <= r < phi-1, or otherwise a(n) = a(q), where quotient q = floor((n+1)/phi) = A005206(n), remainder r = n+1 mod phi, and phi = (1+sqrt(5))/2 = A001622 is the golden ratio.

a(n) = 1 iff A348853(n)+1 mod phi < 2*phi-3, where A348853 is the Zeckendorf "odd" part and the inequality identifies it ending "...101".

a(n) = 0 if n is a Fibonacci number; otherwise a(n) = 1 if n is a Lucas number (A000032); otherwise a(n) = a(A066628(n)) where A066628 discards the most significant Zeckendorf digit.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/phi^2 (A132338). - Amiram Eldar, Feb 17 2024

Example

n = 107 = 1000 1 01 000 in Zeckendorf digits
               ^--- a(107) = 1

Prog

(PARI) { my(phi=quadgen(5), s=phi-1, c=2*phi-3);
a(n) = my(r); until(r<s, [n, r]=divrem(n+1, phi)); r<c; }

Crossrefs

Cf. A189920 and A014417 (Zeckendorf digits), A348853 (odd part).

Cf. A003714 (Fibbinary), A086483.

Cf. A000032, A000045, A001622, A005206, A066628, A132338.

Cf. A038189 (similar binary).

Sequence in context: A354982 A020987 A288372 * A072786 A320926 A144597

Adjacent sequences: A354318 A354319 A354320 * A354322 A354323 A354324

Keyword

nonn,easy

Author

Kevin Ryde, May 29 2022

Status

approved