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A299487 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

%I A299487 #13 Jan 05 2025 19:51:41
%S A299487 4,5,6,7,8,9,10,11,12,13,14,16,17,19,20,22,23,25,26,28,29,31,32,34,35,
%T A299487 37,38,40,41,42,44,45,46,48,49,50,51,53,54,55,57,58,59,60,62,63,64,66,
%U A299487 67,68,69,71,72,73,75,76,77,78,80,81,82,84,85,86,87,89
%N A299487 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.
%C A299487 a(n) = b(n-1) + b(n-2) + b(n-3) for n > 2;
%C A299487 b(0) = least positive integer not in {a(0),a(1),a(2)};
%C A299487 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A299487 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299487 See A022424 for a guide to related sequences.
%H A299487 Clark Kimberling, <a href="/A299487/b299487.txt">Table of n, a(n) for n = 0..1000</a>
%H A299487 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 A299487 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299487 a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6;
%t A299487 a[n_] := a[n] = b[n - 1] + b[n - 2] + b[n - 3];
%t A299487 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299487 u = Table[a[n], {n, 0, 100}]    (* A299486 *)
%t A299487 v = Table[b[n], {n, 0, 100}]    (* A299487 *)
%Y A299487 Cf. A022424, A299486.
%K A299487 nonn,easy
%O A299487 0,1
%A A299487 _Clark Kimberling_, Feb 16 2018