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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.
3
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
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
Sequence in context: A241721 A066491 A282383 * A173334 A331089 A294131
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 16 2018
STATUS
approved