A299534 Solution b( ) of the complementary equation a(n) = b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 2; see Comments.

3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 80, 81, 83, 84, 85, 87, 88

Offset

0,1

Comments

From the Bode-Harborth-Kimberling link:

a(n) = b(n-1) + 2*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 > 0.

Note that (b(n)) is strictly increasing and is the complement of (a(n)).

See A022424 for a guide to related sequences.

Links

Table of n, a(n) for n=0..65.

J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264.

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = b[n - 1] + 2*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A296220 *)
Table[b[n], {n, 0, 100}]    (* A299534 *)

Crossrefs

Cf. A022424, A296220.

Sequence in context: A039237 A039181 A298111 * A026454 A026458 A059109

Adjacent sequences: A299531 A299532 A299533 * A299535 A299536 A299537

Keyword

nonn,easy

Author

Clark Kimberling, Feb 21 2018

Status

approved