A321020 -id:A321020 - cp's OEIS frontend

A156253 Least k such that A054353(k) >= n.

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 47, 48, 49, 50

Offset: 1

Benoit Cloitre, Feb 07 2009

nonn

a(n)=1 plus the number of symbol changes in the first n terms of A000002. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010
From N. J. A. Sloane, Nov 12 2018: (Start)
This seems to be A001462 rewritten so the run lengths are given by A000002. The companion sequence, A000002 rewritten so the run lengths are given by A001462, is A321020.
Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating. (End)
To expand upon N. J. A. Sloane's comments, it's worth noting that Golomb's sequence has a formula from Colin Mallows: g(n) = g(n-g(g(n-1))) + 1, which closely resembles a(n) = a(n-gcd(a(a(n-1)),2)) + 1. - Jon Maiga, May 16 2023

Jon Maiga, Table of n, a(n) for n = 1..10000
J. M. Fedou and G. Fici, Some remarks on differentiable sequences and recursivity, Journal of Integer Sequences 13(3): Article 10.3.2 (2010).
Jon Maiga, A Recurrence Related to the Kolakoski Sequence
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides (Mentions this sequence)

Cf. A000002, A001462, A054353, A156351, A321020.

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 80}, {i, 1, a2[[n]]}]; a3 = Accumulate[a2]; a[1] = 1; a[n_] := a[n] = For[k = a[n - 1], True, k++, If[a3[[k]] >= n, Return[k]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 18 2013 *)
a[1] = 1;
a[n_]:=a[n]=a[n-GCD[a[a[n - 1]], 2]]+1
Array[a, 100] (* Jon Maiga, May 16 2023 *)

Conjecture: a(n) should be asymptotic to 2n/3.
Length of n-th run of the sequence = A000002(n). - Benoit Cloitre, Feb 19 2009
Conjecture: a(n) = (a(a(n-1)) mod 2) + a(n-2) + 1. - Jon Maiga, Dec 09 2021
a(n) = a(n-gcd(a(a(n-1)), 2)) + 1. - Jon Maiga, May 16 2023

A321695 For any sequence f of positive integers, let g(f) be the unique Golomb-like sequence with run lengths given by f and let k(f) be the unique Kolakoski-like sequence with run lengths given by f and initial term 1; this sequence is the unique sequence f satisfying f = g(k(f)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 18, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47

Offset: 1

Rémy Sigrist, Nov 17 2018

nonn

More precisely:
- g(f) is the lexicographically earliest nondecreasing sequence of positive numbers whose RUNS transform equals f,
- k(f) is the lexicographically earliest sequence of 1's and 2's whose RUNS transform equals f,
- in particular:
     g(A001462) = A001462,
     k(A000002) = A000002,
     A321020 = g(A001462).
See A321696 for the RUNS transform of this sequence.
By applying twice the RUNS transform on this sequence, we recover the initial sequence; the same applies for A321696.
This sequence has connections with A288723; in both cases, we have sequences cyclically connected by RUNS transforms.

			We can build this sequence alongside A321696 iteratively:
- this sequence starts with 1,
- hence A321696 starts with 1, 2 (after the initial run of 1's, we have a run of 2's),
- hence this sequence starts with 1, 2, 2, 3 (after the runs of 1's and 2's, we have a run of 3's),
- hence A321696 starts with 1, 2, 2, 1, 1, 2, 2, 2, 1,
- hence this sequence starts 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10,
- etc.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A321695

Cf. A000002, A001462, A288723, A321020, A321696.

PARI
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See Links section.
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A156253 Least k such that A054353(k) >= n.

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