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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. 58

%I

%S 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 72,75,78,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,

%U 129,132,135,138,141,144,147,150,153,156,159,161,164,167,170

%N Solution a( ) of the complementary equation a(n) = 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) = b(n-1) + b(n-2) for n > 2;

%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 ***

%C 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 (a(n)) (b(n)) a(0) a(1)

%C A022424 A055563 1 2

%C A022425 A299407 1 4

%C A022441* A055562 1 5

%C A022426 A299411 2 3

%C A022442* A099467* 2 4

%C A299416 A299417 3 4

%C A299418 A299419 3 5

%C A299420 A299421 4 5

%C A022441* A055562 1 1

%C ***

%C Guide to solutions a( ) and b( ) of a(n) = b(n-1) + b(n-2) + b(n-3) for various initial values:

%C (a(n)) (b(n)) a(0) a(1) a(2)

%C A299486 A299487 1 2 3

%C A299488 A299489 1 2 4

%C A299490 A299491 1 3 5

%C A299492 A299492 2 4 5

%C A299494 A299493 2 4 6

%C A299496 A299494 3 4 5

%C ***

%C Guide to other complementary equations:

%C A022427-A022440: a(n) = b(n-1) + b(n-3)

%C A299531-A299532: a(n) = 2*b(n-1) + b(n-2), a(0) = 1, a(1) = 2

%C A296220, A299534: a(n) = b(n-1) + 2*b(n-2), a(0) = 1, a(1) = 2

%C A022437, A299536: a(n) = b(n-1) + b(n-3), a(0) = 1, a(1) = 2, a(2) = 3

%C A022437, A299538: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 4

%C A022438-A299540: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 5

%C A299541-A299542: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 4, a(2) = 6

%C A299543-A299544: a(n) = 2*b(n-1) + b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3

%C A299545-A299546: a(n) = b(n-1) + 2*b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3

%C A299547: a(n) = b(n-1) + b(n-2) + ... + b(0), a(0) = 1, a(1) = 2, a(2) = 3

%H Ivan Neretin, <a href="/A022424/b022424.txt">Table of n, a(n) for n = 0..10000</a>

%H J-P. Bode, H. Harborth, C. Kimberling, <a href="https://www.fq.math.ca/Papers1/45-3/bode.pdf">Complementary Fibonacci sequences</a>, Fibonacci Quarterly 45 (2007), 254-264.

%t Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 1}]]] &, {1, 2}, Range[56]] (* _Ivan Neretin_, Mar 28 2017 *)

%Y Cf. A055563 (complement), A022425, A299407, A299486-A299494.

%Y Another pair is given in A324142, A324143.

%K nonn

%O 0,2

%A _Clark Kimberling_

%E Edited by _Clark Kimberling_, Feb 16 2018

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Last modified September 20 10:49 EDT 2020. Contains 337264 sequences. (Running on oeis4.)