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A299418
Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 3, a(1) = 5; see Comments.
3
3, 5, 3, 6, 11, 15, 17, 19, 22, 25, 27, 30, 34, 38, 41, 44, 47, 50, 54, 57, 60, 63, 65, 68, 71, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 105, 108, 111, 114, 117, 120, 123, 126, 130, 133, 136, 139, 142, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173
OFFSET
0,1
COMMENTS
a(n) = b(n-1) + b(n-2) for n > 2;
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
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] = 3; a[1] = 5; b[0] = 1; b[1] = 2;
a[n_] := a[n] = 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}] (* A299418 *)
Table[b[n], {n, 0, 100}] (* A299419 *)
CROSSREFS
Sequence in context: A324086 A286108 A096438 * A214456 A118271 A260689
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 15 2018
STATUS
approved