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A377919 a(0) = 0; thereafter a(n) is the lexicographically earliest missing nonnegative integer such that the digits in the sequence (ignoring the commas) alternate in parity.

%I A377919 #32 Dec 09 2024 22:01:27
%S A377919 0,1,2,10,101,21,210,1010,10101,2101,21010,101010,1010101,210101,
%T A377919 2101010,10101010,101010101,21010101,210101010,1010101010,10101010101,
%U A377919 2101010101,21010101010,101010101010,1010101010101,210101010101,2101010101010,10101010101010,101010101010101,21010101010101,210101010101010,1010101010101010,10101010101010101,2101010101010101,21010101010101010,101010101010101010
%N A377919 a(0) = 0; thereafter a(n) is the lexicographically earliest missing nonnegative integer such that the digits in the sequence (ignoring the commas) alternate in parity.
%C A377919 A "lexicographically earliest" variant of A098951.
%H A377919 Paolo Xausa, <a href="/A377919/b377919.txt">Table of n, a(n) for n = 0..1000</a>
%H A377919 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,100,-100,100,-100).
%F A377919 a(0) - a(6) are 0, 1, 2, 10, 101, 21, 210. Thereafter, for k >= 2,
%F A377919   a(4*k-1) = a(4*k-5) || 10,
%F A377919   a(4*k)   = a(4*k-4) || 01,
%F A377919   a(4*k+1) = a(4*k-3) || 01,
%F A377919   a(4*k+2) = a(4*k-2) || 10,
%F A377919 where || denotes concatenation.
%e A377919 We begin by arranging the nonnegative integers whose digits alternate in parity in lexicographic order. First, 0; then the numbers with first digit 1: 1, 10, 101, 1010, 10101, 101010, ...; then the numbers with first digit 2: 2, 20, 201, 2010, 20101, 201010, ...; then the numbers with first digit 3, and so on.
%e A377919 The sequence begins with 0 and from then on we choose the first unused number from the above list which preserves alternating parity of the digits in the sequence.
%e A377919 (The above list does not have an OEIS entry, since there are uncountably many terms before the number 2 appears. In fact there are uncountably many terms before 12 appears. The beginning of the list coincides with A056830.)
%t A377919 LinearRecurrence[{1, -1, 1, 100, -100, 100, -100}, {0, 1, 2, 10, 101, 21, 210, 1010, 10101}, 50] (* _Paolo Xausa_, Dec 09 2024 *)
%Y A377919 Cf. A056830, A098951.
%K A377919 nonn,base
%O A377919 0,3
%A A377919 _N. J. A. Sloane_, Dec 08 2024