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A306211
Concatenation of the current sequence with the lengths of the runs in the sequence, with a(1) = 1.
18
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, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3
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)
CROSSREFS
Positions of 3's, 4's, 5's: A323476, A306222, A306223.
Successive generations: A323477, A323478, A306215, A323475, A306333.
See also A323479, A323480, A323481, A323826 (RUNS transform), A323827, A323829 (where n first appears).
Sequence in context: A269570 A243759 A098398 * A131714 A130196 A230866
KEYWORD
nonn,nice
AUTHOR
A. D. Skovgaard, Jan 29 2019
STATUS
approved