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A319433 Take Zeckendorf representation of n (A014417(n)), drop least significant bit, take inverse Zeckendorf representation.

%I A319433 #45 Jul 29 2025 15:33:06
%S A319433 0,0,1,2,2,3,3,4,5,5,6,7,7,8,8,9,10,10,11,11,12,13,13,14,15,15,16,16,
%T A319433 17,18,18,19,20,20,21,21,22,23,23,24,24,25,26,26,27,28,28,29,29,30,31,
%U A319433 31,32,32,33,34,34,35,36,36,37,37,38,39,39,40,41,41,42,42,43
%N A319433 Take Zeckendorf representation of n (A014417(n)), drop least significant bit, take inverse Zeckendorf representation.
%C A319433 In other words, the Zeckendorf representation of a(n) is obtained from the Zeckendorf representation of n by deleting the least significant bit.
%C A319433 Theorem: The first differences (1,1,0,1,0,1,1,0,...) form the Fibonacci word A005614. (The proof is straightforward.)
%C A319433 The difference sequence agrees with A005614 after the first two terms of A005614. - _Clark Kimberling_, Dec 29 2020
%H A319433 Rémy Sigrist, <a href="/A319433/b319433.txt">Table of n, a(n) for n = 2..25000</a>
%H A319433 Johan Kok, <a href="https://arxiv.org/abs/2507.16500">Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs</a>, arXiv:2507.16500 [math.CO], 2025. See pp. 4-5.
%H A319433 Jeffrey Shallit, <a href="https://arxiv.org/abs/2501.08823">The Hurt-Sada Array and Zeckendorf Representations</a>, arXiv:2501.08823 [math.NT], 2025. See pp. 10-11.
%F A319433 a(n) = floor((n+2)/tau)-1, where tau = golden ratio. - _Clark Kimberling_, Dec 29 2020; corrected by _Harry Altman_, Jun 06 2024
%e A319433 n = 19 has Zeckendorf representation [1, 0, 1, 0, 0, 1], dropping last bit we get [1, 0, 1, 0, 0], which is the Zeckendorf representation of 11, so a(19) = 11.
%t A319433 r = (1 + Sqrt[5])/2; t = Table[Floor[(n - 1)/r] + 2, {n, 0, 150}] (* A319433 *)
%t A319433 Differences[t]  (* A005614 after the 1st 2 terms *)
%t A319433 (* _Clark Kimberling_, Dec 29 2020 *)
%o A319433 (PARI) a(n) = my (f=2, v=0); while (fibonacci(f) < n, f++); while (n > 1, if (n >= fibonacci(f), v += fibonacci(f-1); n -= fibonacci(f); f--); f--); return (v) \\ _Rémy Sigrist_, Oct 04 2018
%Y A319433 Cf. A003714, A005614, A014417.
%K A319433 nonn,base
%O A319433 0,4
%A A319433 _N. J. A. Sloane_, Sep 30 2018
%E A319433 More terms from _Rémy Sigrist_, Oct 04 2018