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A114055 Lexicographically earliest strictly increasing sequence such that the run lengths of digits with equal parity in the sequence's digit stream yield the sequence itself.

%I A114055 #13 Mar 20 2025 12:08:46
%S A114055 1,2,4,5,7,9,10,20,22,31,33,35,36,40,42,44,46,51,53,55,57,59,60,62,64,
%T A114055 66,68,80,82,84,86,88,91,93,95,97,99,111,113,115,117,200,202,204,206,
%U A114055 208,220,222,224,226,228,231,311,313,315,317,319,331,333,335,337
%N A114055 Lexicographically earliest strictly increasing sequence such that the run lengths of digits with equal parity in the sequence's digit stream yield the sequence itself.
%C A114055 A variant of A093848.
%H A114055 Dominic McCarty, <a href="/A114055/b114055.txt">Table of n, a(n) for n = 1..1000</a>
%e A114055 (1),(2,4),(5,7,9,1)(0,20,22),(31,33,35,3)(6,40,42,44,46),(51,53,55,57,59),(60,62,64,66,68,80,82,84,86,88)... Those runs of equal parity digits are of length 1,2,4,5,7,9,10,20... which is the sequence itself.
%o A114055 (Python)
%o A114055 from itertools import groupby
%o A114055 from itertools import count
%o A114055 a = [1,2]
%o A114055 while len(a)<100:a.append(next(k for k in count(a[-1]+1)if (b := [len(list(g))for _,g in(groupby(list(map(lambda d:int(d)%2,"".join(map(str,a))+str(k)))))])and all(b[i]==a[i]for i in range(len(b)-1))and not b[-1]>a[len(b)-1]))
%o A114055 print(a) # _Dominic McCarty_, Mar 18 2025
%Y A114055 Cf. A093848, A101316
%K A114055 base,easy,nonn
%O A114055 1,2
%A A114055 _Eric Angelini_, Feb 02 2006
%E A114055 Name edited, a(40) and on corrected by _Dominic McCarty_, Mar 18 2025