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A022426 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 2, a(1) = 3; see Comments.

%I A022426 #11 Jan 05 2025 19:51:34
%S A022426 2,3,5,10,13,15,17,20,23,26,30,34,37,40,43,46,49,52,55,57,60,63,65,68,
%T A022426 71,74,77,80,83,86,89,92,95,98,101,104,107,110,114,117,120,123,126,
%U A022426 130,133,136
%N A022426 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 2, a(1) = 3; see Comments.
%C A022426 Following the Bode-Harborth-Kimberling link:
%C A022426 a(n) = b(n-1) + b(n-2) for n > 2;
%C A022426 b(0) = least positive integer not in {a(0),a(1)};
%C A022426 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A022426 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A022426 See A022424 for a guide to related sequences.
%H A022426 Ivan Neretin, <a href="/A022426/b022426.txt">Table of n, a(n) for n = 0..10000</a>
%H A022426 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 A022426 Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 1}]]] &, {2, 3}, Range[44]] (* _Ivan Neretin_, Mar 28 2017 *)
%Y A022426 Cf. A022424, A299411 (complement).
%K A022426 nonn
%O A022426 0,1
%A A022426 _Clark Kimberling_