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A299407
Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 1, a(1) = 4; see Comments.
4
2, 3, 6, 7, 8, 10, 11, 12, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 90, 91, 93, 94, 96, 97, 99, 100
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] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
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}] (* A022425 *)
Table[b[n], {n, 0, 100}] (* A299407 *)
CROSSREFS
Sequence in context: A028733 A028789 A038100 * A028757 A184798 A332021
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 14 2018
STATUS
approved