A299531 Solution a( ) of the complementary equation a(n) = 2*b(n-1) + b(n-2), where a(0) = 1, a(1) = 2; see Comments.

1, 2, 11, 14, 17, 20, 23, 26, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 95, 98, 103, 107, 110, 115, 119, 122, 125, 130, 134, 137, 142, 146, 149, 152, 157, 161, 164, 169, 173, 176, 179, 184, 188, 191, 196, 200, 203, 206, 211, 215, 218, 223

Offset

0,2

Comments

From the Bode-Harborth-Kimberling link:

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

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 > 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..56.

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] = 2*b[n - 1] + 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}]    (* A299531 *)
Table[b[n], {n, 0, 100}]    (* A299532 *)

Crossrefs

Cf. A022424, A299532.

Sequence in context: A073914 A266991 A041971 * A106660 A130288 A287395

Adjacent sequences: A299528 A299529 A299530 * A299532 A299533 A299534

Keyword

nonn,easy

Author

Clark Kimberling, Feb 21 2018

Status

approved