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A321695 For any sequence f of positive integers, let g(f) be the unique Golomb-like sequence with run lengths given by f and let k(f) be the unique Kolakoski-like sequence with run lengths given by f and initial term 1; this sequence is the unique sequence f satisfying f = g(k(f)).

%I A321695 #18 May 19 2019 22:05:21
%S A321695 1,2,2,3,3,4,5,6,6,7,7,8,8,9,10,11,12,12,13,13,14,14,15,15,16,17,18,
%T A321695 19,20,21,21,22,22,23,23,24,24,25,25,26,26,27,28,29,30,31,32,33,33,34,
%U A321695 34,35,35,36,36,37,37,38,38,39,39,40,41,42,43,44,45,46,47
%N A321695 For any sequence f of positive integers, let g(f) be the unique Golomb-like sequence with run lengths given by f and let k(f) be the unique Kolakoski-like sequence with run lengths given by f and initial term 1; this sequence is the unique sequence f satisfying f = g(k(f)).
%C A321695 More precisely:
%C A321695 - g(f) is the lexicographically earliest nondecreasing sequence of positive numbers whose RUNS transform equals f,
%C A321695 - k(f) is the lexicographically earliest sequence of 1's and 2's whose RUNS transform equals f,
%C A321695 - in particular:
%C A321695      g(A001462) = A001462,
%C A321695      k(A000002) = A000002,
%C A321695      A321020 = g(A001462).
%C A321695 See A321696 for the RUNS transform of this sequence.
%C A321695 By applying twice the RUNS transform on this sequence, we recover the initial sequence; the same applies for A321696.
%C A321695 This sequence has connections with A288723; in both cases, we have sequences cyclically connected by RUNS transforms.
%H A321695 Rémy Sigrist, <a href="/A321695/b321695.txt">Table of n, a(n) for n = 1..10000</a>
%H A321695 Rémy Sigrist, <a href="/A321695/a321695.gp.txt">PARI program for A321695</a>
%e A321695 We can build this sequence alongside A321696 iteratively:
%e A321695 - this sequence starts with 1,
%e A321695 - hence A321696 starts with 1, 2 (after the initial run of 1's, we have a run of 2's),
%e A321695 - hence this sequence starts with 1, 2, 2, 3 (after the runs of 1's and 2's, we have a run of 3's),
%e A321695 - hence A321696 starts with 1, 2, 2, 1, 1, 2, 2, 2, 1,
%e A321695 - hence this sequence starts 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 10,
%e A321695 - etc.
%o A321695 (PARI) See Links section.
%Y A321695 Cf. A000002, A001462, A288723, A321020, A321696.
%K A321695 nonn
%O A321695 1,2
%A A321695 _Rémy Sigrist_, Nov 17 2018