A299534 - OEIS

%I #4 Feb 22 2018 17:03:29

%S 3,4,5,6,7,8,9,11,12,14,15,17,18,20,21,23,24,26,27,28,30,31,32,33,35,

%T 36,37,39,40,41,42,44,45,46,48,49,50,51,53,54,55,57,58,59,60,62,63,64,

%U 66,67,68,69,71,72,73,75,76,77,78,80,81,83,84,85,87,88

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

%C From the Bode-Harborth-Kimberling link:

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

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

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

%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; b[0] = 3; b[1] = 4;

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

%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}]    (* A296220 *)

%t Table[b[n], {n, 0, 100}]    (* A299534 *)

%Y Cf. A022424, A296220.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Feb 21 2018