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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 20 09:29 EDT 2020. Contains 337264 sequences. (Running on oeis4.)