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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
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
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
Sequence in context: A128460 A273617 A101541 * A059670 A038975 A059456
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 25 2018
STATUS
approved