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A342297 A Van Eck-like sequence based on the Fibonacci numbers. See Comments for definition.

%I A342297 #36 Jul 09 2025 04:55:29
%S A342297 1,1,2,2,3,2,4,5,2,6,5,7,8,2,9,5,7,11,10,12,13,2,14,5,7,16,10,17,13,
%T A342297 15,19,18,20,21,2,22,5,7,24,10,25,13,15,27,18,20,29,23,30,26,28,32,31,
%U A342297 33,34,2,35,5,7,37,10,38,13,15,40,18,20,42,23,43,26,28,45,31,46,34,36,48
%N A342297 A Van Eck-like sequence based on the Fibonacci numbers. See Comments for definition.
%C A342297 We construct finite sequences S_0, S_1, S_2, ...; we let A_i  (i >= 0) be the concatenation of S_0, ..., S_i; and the sequence itself is lim_{i -> oo} A_i. All of S_i, A_i, and A have offset 1. F_i (i >= 0) is the i-th Fibonacci number A000045(i).
%C A342297 S_0 = [1] is special.
%C A342297 For i >= 1, S_i has length F_i, and is defined by:
%C A342297   S_i(j) = max r such that A_{i-1}(r) = j, for 1 <= j <= F_i.
%C A342297 That is, S_i(j) is the index of the most recent occurrence of j in A_{i-1}.
%C A342297 The sequence appears to be as follows. Following the initial 1, it has as a subsequence the Lower Wythoff sequence (A000201: 1, 3, 4, 6, 8, 9, 11, 12, 14, ...) whose terms appear at indices given by the Upper Wythoff sequence (A001950: 2, 5, 7, 10, 13, 15, 18, 20, ...). Interspersed with this is an infinite set of increasingly long initial segments of the Upper Wythoff sequence, each one ending when it has exceeded the most recently appearing term of the Lower Wythoff subsequence. This all shows up clearly on the scatterplot. - _Peter Munn_, Mar 14 2021
%H A342297 Rémy Sigrist, <a href="/A342297/b342297.txt">Table of n, a(n) for n = 1..17711</a>
%H A342297 Rémy Sigrist, <a href="/A342297/a342297.gp.txt">PARI program for A342297</a>
%e A342297 Here are the initial stages of the construction:
%e A342297 S_0 = [1].
%e A342297 A_0 = S_0 = [1].
%e A342297 S_1: When did 1 last appear in A_0? Answer: r=1, S_1 = [1], A_1 = [1,1].
%e A342297 S_2: When did 1 last appear in A_1? Answer: r=2, S_2 = [2], A_2 = [1,1,2].
%e A342297 S_3: When did 1 last appear in A_2? Answer: r=2.
%e A342297 When did 2 last appear in A_2? Answer: r=3. So S_3 = [2,3], A_3 = [1,1,2,2,3].
%e A342297 And so on.
%o A342297 (PARI) See Links section.
%Y A342297 Cf. A000045, A000201, A001950, A181391.
%K A342297 nonn,look
%O A342297 1,3
%A A342297 _N. J. A. Sloane_, Mar 13 2021, following a suggestion from a correspondent who wishes to remain anonymous. The definition given here is my interpretation of his construction