A299531 - OEIS

%I #6 Jan 05 2025 19:51:41

%S 1,2,11,14,17,20,23,26,29,34,38,43,47,52,56,61,65,70,74,79,83,88,92,

%T 95,98,103,107,110,115,119,122,125,130,134,137,142,146,149,152,157,

%U 161,164,169,173,176,179,184,188,191,196,200,203,206,211,215,218,223

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

%C From the Bode-Harborth-Kimberling link:

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

%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 > 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://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 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] = 2*b[n - 1] + 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}]    (* A299531 *)

%t Table[b[n], {n, 0, 100}]    (* A299532 *)

%Y Cf. A022424, A299532.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Feb 21 2018