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

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, 15, 19, 18, 20, 21, 2, 22, 5, 7, 24, 10, 25, 13, 15, 27, 18, 20, 29, 23, 30, 26, 28, 32, 31, 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

Offset: 1

Author

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

Keywords

• nonn
• look

Comments

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).

S_0 = [1] is special.

For i >= 1, S_i has length F_i, and is defined by:

S_i(j) = max r such that A_{i-1}(r) = j, for 1 <= j <= F_i.

That is, S_i(j) is the index of the most recent occurrence of j in A_{i-1}.

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

Examples

			Here are the initial stages of the construction:
S_0 = [1].
A_0 = S_0 = [1].
S_1: When did 1 last appear in A_0? Answer: r=1, S_1 = [1], A_1 = [1,1].
S_2: When did 1 last appear in A_1? Answer: r=2, S_2 = [2], A_2 = [1,1,2].
S_3: When did 1 last appear in A_2? Answer: r=2.
When did 2 last appear in A_2? Answer: r=3. So S_3 = [2,3], A_3 = [1,1,2,2,3].
And so on.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..17711
• Rémy Sigrist, PARI program for A342297

Crossrefs

Cf. A000045, A000201, A001950, A181391.

Programs

PARI

See Links section.