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 A299546 Solution b( ) of the complementary equation a(n) = b(n-1) + 2*b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments. 3

%I

%S 4,5,6,7,8,9,10,11,13,15,17,19,21,23,24,26,27,28,29,31,32,33,35,36,37,

%T 39,40,41,43,44,45,47,48,50,52,53,54,57,59,60,62,63,64,67,68,70,71,72,

%U 75,76,78,79,80,83,84,86,87,88,91,92,94,95,96,98,100,101

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

%C From the Bode-Harborth-Kimberling link:

%C a(n) = b(n-1) + 2*b(n-2) - b(n-3) for n > 3;

%C b(0) = least positive integer not in {a(0),a(1),a(2)};

%C b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.

%C Note that (b(n)) is strictly increasing and is the complement of (a(n)).

%C See A022424 for a guide to related sequences.

%H J-P. Bode, H. Harborth, C. Kimberling, <a href="https://www.fq.math.ca/Papers1/45-3/bode.pdf">Complementary Fibonacci sequences</a>, Fibonacci Quarterly 45 (2007), 254-264.

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5;

%t a[n_] := a[n] = b[n - 1] + 2 b[n - 2] - b[n - 3];

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 100}] (* A299545 *)

%t Table[b[n], {n, 0, 100}] (* A299546 *)

%Y Cf. A022424, A299545.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Mar 01 2018

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)