cp's OEIS Frontend

A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

%I A049999 #40 Jan 05 2025 19:51:36
%S A049999 1,1,1,1,1,1,3,1,1,4,3,1,5,4,1,1,6,5,1,3,7,6,1,1,4,8,7,3,1,5,9,8,4,1,
%T A049999 1,6,10,9,5,1,3,7,11,10,6,1,1,4,8,12,11,7,3,1,5,9,13,12,8,4,1,1,6,10,
%U A049999 14,13,9,5,1,3,7,11,15,14,10,6,1,1,4,8,12,16,15,11
%N A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).
%C A049999 "_David W. Wilson_ conjectured (Dec 14 2005) that" sequence A049998 "consists only of Fibonacci numbers. Proofs were found by _Franklin T. Adams-Watters_ and _Don Reble_, Dec 14 2005." - _Petros Hadjicostas_, Nov 08 2019 [This comment was copied from A049998, which includes _Don Reble_'s proof of the conjecture.]
%H A049999 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-1/quartkimberling01_2004.pdf">Orderings of products of Fibonacci numbers</a>, Fibonacci Quarterly 42:1 (2004), pp. 28-35. (Includes a proof of the conjecture proved in the Comments for sequence A049998.)
%F A049999 A000045(a(n)) = A049998(n) = A049997(n) - A049997(n-1) for n >= 1. - _Petros Hadjicostas_, Nov 08 2019
%e A049999 From _Petros Hadjicostas_, Nov 08 2019: (Start)
%e A049999 A049998(1) = 1 = Fibonacci(1) = Fibonacci(2), so a(1) = min(1,2) = 1.
%e A049999 A049998(7) = 2 = Fibonacci(3), so a(7) = 3.
%e A049999 A049998(10) = 3 = Fibonacci(4), so a(10) = 4.
%e A049999 A049998(13) = 5 = Fibonacci(5), so a(13) = 5.
%e A049999 A049998(17) = 8 = Fibonacci(6), so a(17) = 6. (End)
%Y A049999 Cf. A000045, A049997, A049998, A094563, A226857, A271354.
%K A049999 nonn
%O A049999 1,7
%A A049999 _Clark Kimberling_
%E A049999 Name edited by and more terms from _Petros Hadjicostas_, Nov 08 2019