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A335999
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a(1) = 1; for n >= 2, a(n) = least positive integer not in {a(1),..., a(n-1), b(1),...,b(n-1)}, where for n >=1, b(n) = n + 2 + least positive integer not in {a(1),..., a(n-1), a(n), b(1),...,b(n-1)}.
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2
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1, 2, 3, 4, 6, 8, 10, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 47, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103
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OFFSET
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1,2
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COMMENTS
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In general, let u(1) = 1, and let k be a positive integer. Define u(n) = least
positive integer not in {u(1),..., u(n-1), v(1),...,v(n-1)} and v(n) = n - 1 + k + least positive integer not in {u(1),..., u(n-1), u(n), v(1),...,v(n-1)}. As sets, (u(n)) and (v(n)) are disjoint. If k >=-1, let a(n) = u(n) and b(n) = v(n) for all n >= 1, but if k <= -2, let a(n) = u(n) - k + 1 and b(n) = v(n) - k -1 for all n >= 1. Then every positive integer is in exactly one of the seq (a(n)) and b(n)). The difference sequence of (a(n)) consists of 1's and 2's; the difference sequence of (b(n)) consists of 2's and 3's.
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Guide to related sequences:
k sequences (a(n)) and (b(n))
0 A000201 and A001950 (lower and upper Wythoff sequences)
1 A026351 and A026352
2 A022342 and A003622
3 A335999 and A336008
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LINKS
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Table of n, a(n) for n=1..65.
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EXAMPLE
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a(1) = 1; b(1) = 1+2+2 = 5
a(2) = 2; b(2) = 2+2+3 = 7
a(3) = 3; b(3) = 3+2+4 = 9
a(4) = 4; b(4) = 4+2+6 = 12
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MATHEMATICA
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mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
{a, b} = {{1}, {}}; k = 3;
Do[AppendTo[b, Length[b] + k + mex[Flatten[{a, b}], Last[a]]];
AppendTo[a, mex[Flatten[{a, b}], Last[a]]], {150}]
a (* A335999 *)
b (* A336008 *)
(* Peter J. C. Moses, Jul 13 2020 *)
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CROSSREFS
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Cf. A336008.
Sequence in context: A257457 A122138 A047418 * A026508 A184103 A284038
Adjacent sequences: A335996 A335997 A335998 * A336000 A336001 A336002
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, Jul 16 2020
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STATUS
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approved
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