cp's OEIS Frontend

A299419 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 3, a(1) = 5; see Comments.

%I A299419 #14 Jan 05 2025 19:51:41
%S A299419 1,2,4,7,8,9,10,12,13,14,16,18,20,21,23,24,26,28,29,31,32,33,35,36,37,
%T A299419 39,40,42,43,45,46,48,49,51,52,53,55,56,58,59,61,62,64,66,67,69,70,72,
%U A299419 74,75,77,78,80,81,83,84,86,87,89,90,92,93,95,96,98,99
%N A299419 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 3, a(1) = 5; see Comments.
%C A299419 a(n) = b(n-1) + b(n-2) for n > 2;
%C A299419 b(0) = least positive integer not in {a(0),a(1)};
%C A299419 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A299419 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299419 See A022424 for a guide to related sequences.
%H A299419 Clark Kimberling, <a href="/A299419/b299419.txt">Table of n, a(n) for n = 0..2000</a>
%H A299419 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 A299419 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299419 a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2;
%t A299419 a[n_] := a[n] = b[n - 1] + b[n - 2];
%t A299419 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299419 Table[a[n], {n, 0, 100}]  (* A299418 *)
%t A299419 Table[b[n], {n, 0, 100}]  (* A299419 *)
%Y A299419 Cf. A022424, A299418.
%K A299419 nonn,easy
%O A299419 0,2
%A A299419 _Clark Kimberling_, Feb 16 2018