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%I #8 May 05 2018 04:18:15
%S 3,5,10,13,15,18,20,23,27,30,34,38,40,43,47,50,53,55,59,62,64,68,70,
%T 73,77,79,82,86,89,92,93,99,101,104,108,111,114,115,120,122,126,129,
%U 132,135,140,142,144,146,150,153,157,160,163,165,169,174,176,178
%N Solution (b(n)) of the system of 3 complementary equations in Comments.
%C Define sequences a(n), b(n), c(n) recursively:
%C a(n) = least new;
%C b(n) = least new > = a(n) + n + 1;
%C c(n) = a(n) + b(n);
%C where "least new k" means the least positive integer not yet placed.
%C ***
%C The sequences a,b,c partition the positive integers.
%C ***
%C Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1. (The same limits occur in A298868 and A297838.)
%H Clark Kimberling, <a href="/A299533/b299533.txt">Table of n, a(n) for n = 0..1000</a>
%e n: 0 1 2 3 4 5 6 7 8 9 10
%e a: 1 2 6 8 9 11 12 14 17 19 22
%e b: 3 5 10 13 15 18 20 23 27 30 34
%e c: 4 7 16 21 24 29 32 37 44 49 56
%t z = 200;
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a = {}; b = {}; c = {}; n = 0;
%t Do[{n++;
%t AppendTo[a,
%t mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + n + 1]],
%t AppendTo[c, Last[a] + Last[b]]}, {z}];
%t (* _Peter J. C. Moses_, Apr 23 2018 *)
%t Take[a, 100] (* A297469 *)
%t Take[b, 100] (* A299533 *)
%t Take[c, 100] (* A299423 *)
%t (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y Cf. A299634, A298868, A297838, A297469, A299423.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, May 01 2018
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