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A298418 Solution (c(n)) of the system of 3 complementary equations in Comments.

%I A298418 #12 Jul 27 2024 09:33:04
%S A298418 3,10,13,18,23,31,37,40,43,50,53,63,68,70,73,82,85,89,91,98,111,115,
%T A298418 118,120,129,139,141,144,150,153,155,160,164,168,183,187,189,194,198,
%U A298418 201,203,217,219,232,235,240,247,253,255,258,261,264,268,270,275,284
%N A298418 Solution (c(n)) of the system of 3 complementary equations in Comments.
%C A298418 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298418 a(n) = least new;
%C A298418 b(n) = least new > = a(n) + n + 1;
%C A298418 c(n) = a(n) + b(n);
%C A298418 where "least new k" means the least positive integer not yet placed.
%C A298418 ***
%C A298418 The sequences a,b,c partition the positive integers.
%C A298418 ***
%C A298418 Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C A298418 x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C A298418 x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
%C A298418 (The same limits occur in A298868 and A297469.)
%H A298418 Clark Kimberling, <a href="/A298418/b298418.txt">Table of n, a(n) for n = 0..1000</a>
%e A298418 n:   0   1   2   3   4    5   6   7   8   9  10
%e A298418 a:   1   4   5   7   9   12  15  16  17  20  21
%e A298418 b:   2   6   8  11   14  19  22  24  26  30  32
%e A298418 c:   3  10  13  18   23  31  37  40  43  50  53
%t A298418 z=200;
%t A298418 mex[list_,start_]:=(NestWhile[#+1&,start,MemberQ[list,#]&]);
%t A298418 a={1};b={2};c={3};n=0;
%t A298418 Do[{n++;
%t A298418   AppendTo[a,mex[Flatten[{a,b,c}],If[Length[a]==0,1,Last[a]]]],
%t A298418   AppendTo[b,mex[Flatten[{a,b,c}],Last[a]+n+1]],
%t A298418   AppendTo[c,Last[a]+Last[b]]},{z}];
%t A298418 Take[a,100] (* A297838 *)
%t A298418 Take[b,100] (* A298170 *)
%t A298418 Take[c,100] (* A298418 *)
%t A298418 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A298418 Cf. A299634, A298868, A297469, A297838, A298170.
%K A298418 nonn,easy
%O A298418 0,1
%A A298418 _Clark Kimberling_, May 01 2018