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A022424 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 1, a(1) = 2; see Comments.

%I A022424 #42 Jan 05 2025 19:51:34
%S A022424 1,2,7,9,11,14,18,22,25,28,31,33,36,39,41,44,47,50,53,56,59,62,66,69,
%T A022424 72,75,78,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,
%U A022424 129,132,135,138,141,144,147,150,153,156,159,161,164,167,170
%N A022424 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 1, a(1) = 2; see Comments.
%C A022424 From the Bode-Harborth-Kimberling link:
%C A022424 a(n) = b(n-1) + b(n-2) for n > 2;
%C A022424 b(0) = least positive integer not in {a(0),a(1)};
%C A022424 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A022424 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A022424 ***
%C A022424 In the following guide to solutions a( ) and b( ) of a(n) = b(n-1) + b(n-2), an asterisk (*) indicates that a( ) differs from the indicated A-sequence in one or two initial terms:
%C A022424 (a(n))     (b(n))    a(0)   a(1)
%C A022424 A022424    A055563     1      2
%C A022424 A022425    A299407     1      4
%C A022424 A022441*   A055562     1      5
%C A022424 A022426    A299411     2      3
%C A022424 A022442*   A099467*    2      4
%C A022424 A299416    A299417     3      4
%C A022424 A299418    A299419     3      5
%C A022424 A299420    A299421     4      5
%C A022424 A022441*   A055562     1      1
%C A022424 ***
%C A022424 Guide to solutions a( ) and b( ) of a(n) = b(n-1) + b(n-2) + b(n-3) for various initial values:
%C A022424 (a(n))     (b(n))    a(0)   a(1)   a(2)
%C A022424 A299486    A299487     1      2      3
%C A022424 A299488    A299489     1      2      4
%C A022424 A299490    A299491     1      3      5
%C A022424 A299492    A299492     2      4      5
%C A022424 A299494    A299493     2      4      6
%C A022424 A299496    A299494     3      4      5
%C A022424 ***
%C A022424 Guide to other complementary equations:
%C A022424 A022427-A022440:  a(n) = b(n-1) + b(n-3)
%C A022424 A299531-A299532:  a(n) = 2*b(n-1) + b(n-2), a(0) = 1, a(1) = 2
%C A022424 A296220, A299534: a(n) = b(n-1) + 2*b(n-2), a(0) = 1, a(1) = 2
%C A022424 A022437, A299536: a(n) = b(n-1) + b(n-3), a(0) = 1, a(1) = 2, a(2) = 3
%C A022424 A022437, A299538: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 4
%C A022424 A022438-A299540:  a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 5
%C A022424 A299541-A299542:  a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 4, a(2) = 6
%C A022424 A299543-A299544:  a(n) = 2*b(n-1) + b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3
%C A022424 A299545-A299546:  a(n) = b(n-1) + 2*b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3
%C A022424 A299547: a(n) = b(n-1) + b(n-2) + ... + b(0), a(0) = 1, a(1) = 2, a(2) = 3
%H A022424 Ivan Neretin, <a href="/A022424/b022424.txt">Table of n, a(n) for n = 0..10000</a>
%H A022424 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 A022424 Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 1}]]] &, {1, 2}, Range[56]] (* _Ivan Neretin_, Mar 28 2017 *)
%Y A022424 Cf. A055563 (complement), A022425, A299407, A299486-A299494.
%Y A022424 Another pair is given in A324142, A324143.
%K A022424 nonn
%O A022424 0,2
%A A022424 _Clark Kimberling_
%E A022424 Edited by _Clark Kimberling_, Feb 16 2018