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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-François 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.)