# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a299420 Showing 1-1 of 1 %I A299420 #12 Mar 05 2018 13:46:12 %S A299420 4,5,3,8,13,16,19,21,23,26,29,32,35,38,42,46,49,52,55,58,61,64,67,70, %T A299420 73,76,79,81,84,87,89,92,95,98,101,104,107,110,113,116,119,122,125, %U A299420 128,131,134,137,140,143,146,149,152,155,158,162,165,168,171,174 %N A299420 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 4, a(1) = 5; see Comments. %C A299420 a(n) = b(n-1) + b(n-2) for n > 2; %C A299420 b(0) = least positive integer not in {a(0),a(1)}; %C A299420 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1. %C A299420 Note that (b(n)) is strictly increasing and is the complement of (a(n)). %C A299420 See A022424 for a guide to related sequences. %H A299420 Clark Kimberling, Table of n, a(n) for n = 0..2000 %H A299420 J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264. %t A299420 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; %t A299420 a[0] = 4; a[1] = 5; b[0] = 1; b[1] = 2; %t A299420 a[n_] := a[n] = b[n - 1] + b[n - 2]; %t A299420 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; %t A299420 Table[a[n], {n, 0, 100}] (* A299420 *) %t A299420 Table[b[n], {n, 0, 100}] (* A299421 *) %Y A299420 Cf. A022424, A299421. %K A299420 nonn,easy %O A299420 0,1 %A A299420 _Clark Kimberling_, Feb 16 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE