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A299538
Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 3, a(2) = 4; see Comments.
2
1, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 19, 21, 23, 24, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 101
OFFSET
0,2
COMMENTS
From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + 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] = 2; a[1] = 3; a[2] = 4; b[0] = 1; b[1] = 5;
a[n_] := a[n] = b[n - 1] + 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}] (* A022437 *)
Table[b[n], {n, 0, 100}] (* A299538 *)
CROSSREFS
Sequence in context: A205705 A343616 A270653 * A201512 A242290 A296562
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 25 2018
STATUS
approved