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A298170
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Solution (b(n)) of the system of 3 complementary equations in Comments.
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3
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2, 6, 8, 11, 14, 19, 22, 24, 26, 30, 32, 38, 41, 42, 44, 49, 51, 54, 55, 59, 66, 69, 71, 72, 77, 83, 84, 86, 90, 92, 93, 96, 99, 101, 109, 112, 113, 116, 119, 121, 122, 130, 131, 138, 140, 143, 147, 151, 152, 154, 156, 158, 161, 162, 165, 170, 174, 181, 184
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refs;
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history;
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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) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = least new > = a(n) + n + 1;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
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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 10
a: 1 4 5 7 9 12 15 16 17 20 21
b: 2 6 8 11 14 19 22 24 26 30 32
c: 3 10 13 18 23 31 37 40 43 50 53
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MATHEMATICA
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z=200;
mex[list_, start_]:=(NestWhile[#+1&, start, MemberQ[list, #]&]);
a={1}; b={2}; c={3}; n=0;
Do[{n++;
AppendTo[a, mex[Flatten[{a, b, c}], If[Length[a]==0, 1, Last[a]]]],
AppendTo[b, mex[Flatten[{a, b, c}], Last[a]+n+1]],
AppendTo[c, Last[a]+Last[b]]}, {z}];
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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