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A297465
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Solution (b(n)) of the system of 4 complementary equations in Comments.
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3
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2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 35, 39, 42, 45, 49, 52, 55, 59, 62, 65, 69, 72, 76, 79, 82, 85, 89, 92, 95, 99, 102, 105, 109, 112, 115, 119, 122, 125, 129, 132, 135, 139, 142, 145, 149, 152, 155, 159, 162, 166, 169, 172, 175, 179, 182, 185, 189, 192
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OFFSET
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0,1
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COMMENTS
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Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:
a(n) = least new;
b(n) = least new;
c(n) = least new;
d(n) = a(n) + b(n) + c(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 10n - 6 - 3 a(n) <= 2
0 <= 10n - 2 - 3 b(n) <= 3
0 <= 10n +1 - 3 c(n) <= 3
0 <= 10n - 3 - d(n) <= 2
***
The sequences a,b,c,d partition the positive integers. The sequence d can be called the "anti-tribonacci sequence"; viz., if sequences a and b are defined as above, and c(n) is defined by c(n) = a(n) + b(n), then the resulting system of 3 complementary sequences gives c = A036554, the "anti-Fibonacci sequence."
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LINKS
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EXAMPLE
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n: 0 1 2 3 4 5 6 7 8 9
a: 1 4 8 11 14 18 21 24 28 31
b: 2 5 9 12 15 19 22 25 29 32
c: 3 7 10 13 17 20 23 26 30 33
d: 6 16 27 36 46 57 66 75 87 96
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MATHEMATICA
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z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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