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A299545
Solution a( ) of the complementary equation a(n) = b(n-1) + 2*b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.
3
1, 2, 3, 12, 14, 16, 18, 20, 22, 25, 30, 34, 38, 42, 46, 49, 51, 55, 56, 58, 61, 65, 66, 69, 73, 74, 77, 81, 82, 85, 89, 90, 93, 97, 99, 104, 107, 108, 112, 119, 121, 123, 127, 128, 132, 138, 139, 143, 144, 148, 154, 155, 159, 160, 164, 170, 171, 175, 176
OFFSET
0,2
COMMENTS
From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + 2*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] = b[n - 1] + 2 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}] (* A299545 *)
Table[b[n], {n, 0, 100}] (* A299546 *)
CROSSREFS
Sequence in context: A039588 A024579 A291565 * A068603 A045878 A295398
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 01 2018
STATUS
approved