A299488 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 2, a(2) = 4; see Comments.

1, 2, 4, 14, 18, 21, 24, 27, 30, 33, 36, 40, 44, 48, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 177, 181, 185, 189, 193, 197, 201, 204, 208, 212, 216, 220, 224

Offset

0,2

Comments

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

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

Clark Kimberling, Table of n, a(n) for n = 0..1000

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] = 4; b[0] = 3; b[1] = 5; b[2] = 6;
a[n_] := a[n] = 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}]    (* A299488 *)
Table[b[n], {n, 0, 100}]    (* A299489 *)

Crossrefs

Cf. A022424, A299489.

Sequence in context: A360713 A253444 A336821 * A295401 A342605 A032398

Adjacent sequences: A299485 A299486 A299487 * A299489 A299490 A299491

Keyword

nonn,easy,changed

Author

Clark Kimberling, Feb 16 2018

Status

approved