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A297466
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Solution (c(n)) of the system of 4 complementary equations in Comments.
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3
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3, 7, 10, 13, 17, 20, 23, 26, 30, 33, 37, 40, 43, 47, 50, 53, 56, 60, 63, 67, 70, 73, 77, 80, 83, 86, 90, 93, 97, 100, 103, 107, 110, 113, 116, 120, 123, 127, 130, 133, 137, 140, 143, 146, 150, 153, 157, 160, 163, 167, 170, 173, 176, 180, 183, 187, 190, 193
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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."
Linear recurrence with signature (1,0,0,0,0,0,0,0,1,-1).
G.f.: (3 + 4*x + 3*x^2 + 3*x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8)/(1 - x - x^9 + x^10). (End)
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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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