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A299420 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 4, a(1) = 5; see Comments.

%I A299420 #13 Jan 05 2025 19:51:41
%S A299420 4,5,3,8,13,16,19,21,23,26,29,32,35,38,42,46,49,52,55,58,61,64,67,70,
%T A299420 73,76,79,81,84,87,89,92,95,98,101,104,107,110,113,116,119,122,125,
%U A299420 128,131,134,137,140,143,146,149,152,155,158,162,165,168,171,174
%N A299420 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 4, a(1) = 5; see Comments.
%C A299420 a(n) = b(n-1) + b(n-2) for n > 2;
%C A299420 b(0) = least positive integer not in {a(0),a(1)};
%C A299420 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A299420 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299420 See A022424 for a guide to related sequences.
%H A299420 Clark Kimberling, <a href="/A299420/b299420.txt">Table of n, a(n) for n = 0..2000</a>
%H A299420 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 A299420 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299420 a[0] = 4; a[1] = 5; b[0] = 1; b[1] = 2;
%t A299420 a[n_] := a[n] = b[n - 1] + b[n - 2];
%t A299420 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299420 Table[a[n], {n, 0, 100}]  (* A299420 *)
%t A299420 Table[b[n], {n, 0, 100}]  (* A299421 *)
%Y A299420 Cf. A022424, A299421.
%K A299420 nonn,easy
%O A299420 0,1
%A A299420 _Clark Kimberling_, Feb 16 2018