OFFSET

1,3

COMMENTS

Conjecture: All terms are less than or equal to 5. - Peter Kagey, Jan 29 2019

Conjecture: Every number appears! (Based on the analogy with the somewhat similar sequence A090822, where the first 5 appeared at around 10^(10^23) steps). - N. J. A. Sloane, Jan 29 2019

An alternative definition: Start with 1, extend the sequence by appending its RUNS transform, recompute the RUNS transform, append it, repeat. - N. J. A. Sloane, Jan 29 2019

The first time we see 1, 2, 3, 4, 5 is at n=1, 3, 37, 60, 255. After 65 generations (10228800161220 terms) the largest term is 5. The relative frequencies of 1..5 are roughly 0.71, 6.7e-9, 0.23, 1.6e-8, 0.061. 2s and 4s appear to get rarer as n increases. - Benjamin Chaffin, Feb 07 2019

If we record the successive RUNS transforms and concatenate them, we get 1; 2; 2, 1; 2, 2, 1; 2, 2, 1, 2, 1; ..., which is this sequence without the initial 1. - A. D. Skovgaard, Jan 30 2019 (Rephrased by N. J. A. Sloane, Jan 30 2019)

LINKS

Peter Kagey, Table of n, a(n) for n = 1..10029 (first 20 generations)

N. J. A. Sloane, Table of n, a(n) for n = 1..236878 (first 27 generations)

N. J. A. Sloane, Notes on A306211, Feb 01 2019

EXAMPLE

a(2) = 1, since there is a run of length 1 at a(1).

a(3) = 2, since there is a run of length 2 at a(1..2).

a(4..5) = 2, 1, since the runs are as follows:

1, 1, 2 a(1..3)

\__/ |

2, 1 a(4..5)

a(37) = 3, since a(20..22) = 1, 1, 1.

Steps in construction:

[1] initial sequence

[1] its run length

.

[1, 1] concatenation of above is new sequence

[2] its run length

.

[1, 1, 2] concatenation of above is new sequence

[2, 1] its run lengths

.

[1, 1, 2, 2, 1]

[2, 2, 1]

.

[1, 1, 2, 2, 1, 2, 2, 1]

[2, 2, 1, 2, 1]

.

[1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1]

[2, 2, 1, 2, 1, 2, 1, 1, 1]

.

[1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1]

[2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]

.

[1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]

From N. J. A. Sloane, Jan 31 2019: (Start)

The first 9 generations, in compressed notation (see A323477) are:

1

11

112

11221

11221221

1122122122121

1122122122121221212111

1122122122121221212111221212111211113

1122122122121221212111221212111211113221212111211113211113141

... (End)

MATHEMATICA

seq[n_] := seq[n] = If[n==1, {1}, Join[seq[n-1], Length /@ Split[seq[n-1]]]];

seq[10] (* Jean-François Alcover, Jul 19 2022 *)

PROG

(Haskell)

group [] = []

group (x:xs)= (x:ys):group zs where (ys, zs) = span (==x) xs

a306211_next_gen xs = xs ++ (map length $ group xs)

a306211_gen 1 = [1]

a306211_gen n = a306211_next_gen $ a306211_gen (n-1)

a306211 n = a306211_gen n !! (n-1)

-- Jean-François Antoniotti, Jan 31 2021

CROSSREFS

KEYWORD

nonn,nice

AUTHOR

A. D. Skovgaard, Jan 29 2019

STATUS

approved