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A299411 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 2, a(1) = 3; see Comments.

%I A299411 #10 Jan 05 2025 19:51:41
%S A299411 1,4,6,7,8,9,11,12,14,16,18,19,21,22,24,25,27,28,29,31,32,33,35,36,38,
%T A299411 39,41,42,44,45,47,48,50,51,53,54,56,58,59,61,62,64,66,67,69,70,72,73,
%U A299411 75,76,78,79,81,82,84,85,87,88,90,91,93,94,96,97,99,100
%N A299411 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 2, a(1) = 3; see Comments.
%C A299411 a(n) = b(n-1) + b(n-2) for n > 2;
%C A299411 b(0) = least positive integer not in {a(0),a(1)};
%C A299411 b(n) = least positive integer not in {a(0),...,a(n),b(0),...b(n-1)} for n > 1.
%C A299411 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299411 See A022424 for a guide to related sequences.
%H A299411 Clark Kimberling, <a href="/A299411/b299411.txt">Table of n, a(n) for n = 0..2000</a>
%H A299411 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 A299411 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299411 a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
%t A299411 a[n_] := a[n] = b[n - 1] + b[n - 2];
%t A299411 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299411 Table[a[n], {n, 0, 100}]  (* A022426 *)
%t A299411 Table[b[n], {n, 0, 100}]  (* A299411 *)
%Y A299411 Cf. A022424, A022426.
%K A299411 nonn,easy
%O A299411 0,2
%A A299411 _Clark Kimberling_, Feb 14 2018