A299486 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) = 3; see Comments.

1, 2, 3, 15, 18, 21, 24, 27, 30, 33, 36, 39, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 110, 115, 119, 123, 127, 131, 135, 139, 143, 147, 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] = 3; b[0] = 4; 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}]]];
u = Table[a[n], {n, 0, 100}]    (* A299486 *)
v = Table[b[n], {n, 0, 100}]    (* A299487 *)

Crossrefs

Cf. A022424, A299487.

Sequence in context: A241721 A066491 A282383 * A173334 A331089 A294131

Adjacent sequences: A299483 A299484 A299485 * A299487 A299488 A299489

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2018

Status

approved