A299543 Solution a( ) of the complementary equation a(n) = 2*b(n-1) + b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

1, 2, 3, 13, 15, 17, 19, 21, 23, 25, 29, 34, 38, 42, 46, 50, 54, 56, 57, 61, 64, 65, 67, 71, 74, 75, 79, 82, 83, 87, 90, 91, 95, 98, 99, 103, 106, 107, 111, 118, 121, 121, 125, 128, 133, 139, 140, 141, 145, 148, 153, 157, 157, 161, 164, 169, 173, 173, 177

Offset

0,2

Comments

From the Bode-Harborth-Kimberling link:

a(n) = 2*b(n-1) + 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] = 2 b[n - 1] + 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}]    (* A299543 *)
Table[b[n], {n, 0, 100}]    (* A299544 *)

Crossrefs

Cf. A022424, A299544.

Sequence in context: A128460 A273617 A101541 * A059670 A038975 A059456

Adjacent sequences: A299540 A299541 A299542 * A299544 A299545 A299546

Keyword

nonn,easy

Author

Clark Kimberling, Feb 25 2018

Status

approved