

A306211


Concatenation of the current sequence with the lengths of the runs in the sequence, with a(1) = 1.


16



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
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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, 6.7e9, 0.23, 1.6e8, 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)


MAPLE

P:=proc(q) local a, b, c, k, n; a:=[1, 1];
for n from 1 to q do b:=1; c:=[];
for k from 1 to nops(a)1 do if a[k+1]=a[k] then b:=b+1;
else c:=[op(c), b]; b:=1; fi; od; a:=[op(a), op(c), b]; od;
a; end: P(10); # Paolo P. Lava, Jan 30 2019. P(g) produces generations 1 through g+2.


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 (n1)
a306211 n = a306211_gen n !! (n1)
 JeanFrançois Antoniotti, Jan 31 2021


CROSSREFS

Cf. A000002, A107946, A306215, A090822.
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
Adjacent sequences: A306208 A306209 A306210 * A306212 A306213 A306214


KEYWORD

nonn,nice


AUTHOR

A. D. Skovgaard, Jan 29 2019


STATUS

approved



