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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 (list; graph; refs; listen; history; text; internal format)
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)

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 (n-1)

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

-- Jean-Fran├žois 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

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)