A022442 -id:A022442 - cp's OEIS frontend

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

1, 2, 7, 9, 11, 14, 18, 22, 25, 28, 31, 33, 36, 39, 41, 44, 47, 50, 53, 56, 59, 62, 66, 69, 72, 75, 78, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 161, 164, 167, 170

Offset: 0

Clark Kimberling

nonn

From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + b(n-2) for n > 2;
b(0) = least positive integer not in {a(0),a(1)};
b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
Note that (b(n)) is strictly increasing and is the complement of (a(n)).
***
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:
(a(n))     (b(n))    a(0)   a(1)
A022424    A055563     1      2
A022425    A299407     1      4
A022441*   A055562     1      5
A022426    A299411     2      3
A022442*   A099467*    2      4
A299416    A299417     3      4
A299418    A299419     3      5
A299420    A299421     4      5
A022441*   A055562     1      1
***
Guide to solutions a( ) and b( ) of a(n) = b(n-1) + b(n-2) + b(n-3) for various initial values:
(a(n))     (b(n))    a(0)   a(1)   a(2)
A299486    A299487     1      2      3
A299488    A299489     1      2      4
A299490    A299491     1      3      5
A299492    A299492     2      4      5
A299494    A299493     2      4      6
A299496    A299494     3      4      5
***
Guide to other complementary equations:
A022427-A022440:  a(n) = b(n-1) + b(n-3)
A299531-A299532:  a(n) = 2*b(n-1) + b(n-2), a(0) = 1, a(1) = 2
A296220, A299534: a(n) = b(n-1) + 2*b(n-2), a(0) = 1, a(1) = 2
A022437, A299536: a(n) = b(n-1) + b(n-3), a(0) = 1, a(1) = 2, a(2) = 3
A022437, A299538: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 4
A022438-A299540:  a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 5
A299541-A299542:  a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 4, a(2) = 6
A299543-A299544:  a(n) = 2*b(n-1) + b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3
A299545-A299546:  a(n) = b(n-1) + 2*b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3
A299547: a(n) = b(n-1) + b(n-2) + ... + b(0), a(0) = 1, a(1) = 2, a(2) = 3

Ivan Neretin, Table of n, a(n) for n = 0..10000
J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264.

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

Another pair is given in A324142, A324143.

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

Edited by Clark Kimberling, Feb 16 2018

A099467 a(1) = a(2) = 1; for n > 2, a(n) is the smallest number > a(n-1) which is not the sum of 2 consecutive elements of the sequence.

Original entry on oeis.org

1, 1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 97

Offset: 1

Gaetan Polard (gaetan27(AT)hotmail.com), Nov 18 2004

The first differences are 1 and 2 strictly alternately, except near powers of 2: a(2^k+2)-a(2^k+1) = a(2^k+1)-a(2^k). Cf. A001651 which is generated by the same rule if we start from 1, 2 and has first differences 1, 2, 1, 2... with no exceptions. - Andrey Zabolotskiy, Feb 11 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A022442 (complement), A001651.

A[1]:= 1: A[2]:= 1: forbid:= {2}:
for n from 3 to 100 do
  for k from A[n-1]+1 while member(k, forbid) do od:
  A[n]:= k;
  forbid:= forbid union {A[n-1]+k};
od:
seq(A[i],i=1..100); # Robert Israel, Nov 29 2017

cp's OEIS Frontend

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

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