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A299541 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 4, a(2) = 6; see Comments.

%I A299541 #5 Jan 05 2025 19:51:41
%S A299541 2,4,6,6,10,13,16,19,21,25,27,31,33,37,40,43,46,49,52,55,58,61,64,67,
%T A299541 70,73,75,79,81,85,87,91,93,97,99,103,105,109,111,115,117,121,123,127,
%U A299541 129,133,135,139,141,145,148,151,154,157,160,163,166,169,172
%N A299541 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 4, a(2) = 6; see Comments.
%C A299541 From the Bode-Harborth-Kimberling link:
%C A299541 a(n) = b(n-1) + b(n-3) for n > 3;
%C A299541 b(0) = least positive integer not in {a(0),a(1),a(2)};
%C A299541 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A299541 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299541 See A022424 for a guide to related sequences.
%H A299541 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 A299541 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299541 a[0] = 2; a[1] = 4; a[2] = 6; b[0] = 1; b[1] = 3;
%t A299541 a[n_] := a[n] = b[n - 1] + b[n - 3];
%t A299541 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299541 Table[a[n], {n, 0, 100}]    (* A299541 *)
%t A299541 Table[b[n], {n, 0, 100}]    (* A299542 *)
%Y A299541 Cf. A022424, A299542.
%K A299541 nonn,easy
%O A299541 0,1
%A A299541 _Clark Kimberling_, Feb 25 2018