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A299546
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Solution b( ) 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.
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3
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4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 52, 53, 54, 57, 59, 60, 62, 63, 64, 67, 68, 70, 71, 72, 75, 76, 78, 79, 80, 83, 84, 86, 87, 88, 91, 92, 94, 95, 96, 98, 100, 101
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..65.
J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A022424, A299545.
Sequence in context: A039174 A016069 A194283 * A039128 A214421 A294237
Adjacent sequences: A299543 A299544 A299545 * A299547 A299548 A299549
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Mar 01 2018
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STATUS
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approved
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