A299545 Solution a( ) of the complementary equation a(n) = b(n-1) + 2*b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

1, 2, 3, 12, 14, 16, 18, 20, 22, 25, 30, 34, 38, 42, 46, 49, 51, 55, 56, 58, 61, 65, 66, 69, 73, 74, 77, 81, 82, 85, 89, 90, 93, 97, 99, 104, 107, 108, 112, 119, 121, 123, 127, 128, 132, 138, 139, 143, 144, 148, 154, 155, 159, 160, 164, 170, 171, 175, 176

Offset

0,2

Comments

From the Bode-Harborth-Kimberling link:

a(n) = b(n-1) + 2*b(n-2) - b(n-3) for n > 3;

b(0) = least positive integer not in {a(0),a(1),a(2)};

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)).

See A022424 for a guide to related sequences.

Links

Table of n, a(n) for n=0..58.

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; a[2] = 3; b[0] = 4; b[1] = 5;
a[n_] := a[n] = b[n - 1] + 2 b[n - 2] - b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299545 *)
Table[b[n], {n, 0, 100}]    (* A299546 *)

Crossrefs

Cf. A022424, A299546.

Sequence in context: A039588 A024579 A291565 * A068603 A045878 A295398

Adjacent sequences: A299542 A299543 A299544 * A299546 A299547 A299548

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2018

Status

approved