A299547 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2) + ... + b(0), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

1, 2, 3, 12, 18, 25, 33, 42, 52, 63, 76, 90, 105, 121, 138, 157, 177, 198, 220, 243, 267, 293, 320, 348, 377, 407, 438, 470, 504, 539, 575, 612, 650, 689, 729, 770, 813, 857, 902, 948, 995, 1043, 1092, 1142, 1193, 1246, 1300, 1355, 1411, 1468, 1526, 1585

Offset

0,2

Comments

From the Bode-Harborth-Kimberling link:

a(n) = b(n-1) + b(n-2) + ... + b(0) 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..51.

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;
a[n_] := a[n] = Sum[b[k], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299547 *)

Crossrefs

Cf. A022424.

Sequence in context: A112976 A100570 A056700 * A140989 A281086 A355847

Adjacent sequences: A299544 A299545 A299546 * A299548 A299549 A299550

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2018

Extensions

Definition corrected by Georg Fischer, Sep 28 2020

Status

approved