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A299421 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 4, a(1) = 5; see Comments.

%I A299421 #13 Jan 05 2025 19:51:41
%S A299421 1,2,6,7,9,10,11,12,14,15,17,18,20,22,24,25,27,28,30,31,33,34,36,37,
%T A299421 39,40,41,43,44,45,47,48,50,51,53,54,56,57,59,60,62,63,65,66,68,69,71,
%U A299421 72,74,75,77,78,80,82,83,85,86,88,90,91,93,94,96,97,99,100
%N A299421 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 4, a(1) = 5; see Comments.
%C A299421 a(n) = b(n-1) + b(n-2) for n > 2;
%C A299421 b(0) = least positive integer not in {a(0),a(1)};
%C A299421 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A299421 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299421 See A022424 for a guide to related sequences.
%H A299421 Clark Kimberling, <a href="/A299421/b299421.txt">Table of n, a(n) for n = 0..2000</a>
%H A299421 J-P. Bode, H. Harborth, C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-3/bode.pdf">Complementary Fibonacci sequences</a>, Fibonacci Quarterly 45 (2007), 254-264.
%t A299421 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299421 a[0] = 4; a[1] = 5; b[0] = 1; b[1] = 2;
%t A299421 a[n_] := a[n] = b[n - 1] + b[n - 2];
%t A299421 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299421 Table[a[n], {n, 0, 100}]  (* A299420 *)
%t A299421 Table[b[n], {n, 0, 100}]  (* A299421 *)
%Y A299421 Cf. A022424, A299420.
%K A299421 nonn,easy
%O A299421 0,2
%A A299421 _Clark Kimberling_, Feb 16 2018