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Solution (c(n)) of the system of 3 complementary equations in Comments.

3

`%I #8 May 05 2018 04:18:27
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`%S 3,9,16,19,22,28,36,41,48,57,61,66,74,77,83,89,94,97,101,103,108,115,
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`%T 121,130,133,136,139,146,154,157,161,166,171,178,183,191,200,209,214,
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`%U 217,222,229,238,241,244,248,253,257,265,275,282,290,295,298,306,317
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`%N Solution (c(n)) of the system of 3 complementary equations in Comments.
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`%C Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
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`%C a(n) = least new k >= 2*b(n-1);
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`%C b(n) = least new k;
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`%C c(n) = a(n) + b(n);
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`%C where "least new k" means the least positive integer not yet placed.
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`%C ***
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`%C The sequences a,b,c partition the positive integers.
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`%C ***
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`%C Let x = 11/6. Conjectures:
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`%C a(n) - 2*n*x = 0 for infinitely many n;
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`%C b(n) - n*x = 0 for infinitely many n;
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`%C c(n) - 3*n*x = 0 for infinitely many n;
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`%C (a(n) - 2*n*x) is unbounded below and above;
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`%C (b(n) - n*x) is unbounded below and above;
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`%C (c(n) - 3*n*x) is unbounded below and above;
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`%C ***
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`%C Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
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`%C 12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
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`%C 2 occurs infinitely many times in d(c).
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`%H Clark Kimberling, <a href="/A299636/b299636.txt">Table of n, a(n) for n = 0..1000</a>
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`%e n: 0 1 2 3 4 5 6 7 8 9
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`%e a: 1 4 10 12 14 17 23 26 30 37
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`%e b: 2 5 6 7 8 11 13 15 18 20
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`%e c: 3 9 16 19 22 28 36 41 48 57
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`%t z = 1000;
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`%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
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`%t a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
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`%t Do[{
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`%t AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
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`%t AppendTo[b, mex[Flatten[{a, b, c}], 1]],
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`%t AppendTo[c, Last[a] + Last[b]]}, {z}];
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`%t Take[a, 100] (* A299634 *)
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`%t Take[b, 100] (* A299635 *)
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`%t Take[c, 100] (* A299636 *)
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`%t (* _Peter J. C. Moses_, Apr 08 2018 *)
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`%Y Cf. A299634, A299635.
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`%K nonn,easy
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`%O 0,1
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`%A _Clark Kimberling_, Apr 17 2018
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