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A297838 Solution (a(n)) of the system of 3 complementary equations in Comments.

%I A297838 #11 May 01 2018 03:01:05
%S A297838 1,4,5,7,9,12,15,16,17,20,21,25,27,28,29,33,34,35,36,39,45,46,47,48,
%T A297838 52,56,57,58,60,61,62,64,65,67,74,75,76,78,79,80,81,87,88,94,95,97,
%U A297838 100,102,103,104,105,106,107,108,110,114,117,123,124,125,126,127
%N A297838 Solution (a(n)) of the system of 3 complementary equations in Comments.
%C A297838 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A297838 a(n) = least new;
%C A297838 b(n) = least new > = a(n) + n + 1;
%C A297838 c(n) = a(n) + b(n);
%C A297838 where "least new k" means the least positive integer not yet placed.
%C A297838 ***
%C A297838 The sequences a,b,c partition the positive integers.
%C A297838 ***
%C A297838 Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C A297838 x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C A297838 x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
%C A297838 (The same limits occur in A298868 and A297469.)
%H A297838 Clark Kimberling, <a href="/A297838/b297838.txt">Table of n, a(n) for n = 0..1000</a>
%e A297838 n:   0   1   2   3   4    5   6   7   8   9  10
%e A297838 a:   1   4   5   7   9   12  15  16  17  20  21
%e A297838 b:   2   6   8  11   14  19  22  24  26  30  32
%e A297838 c:   3  10  13  18   23  31  37  40  43  50  53
%t A297838 z=200;
%t A297838 mex[list_,start_]:=(NestWhile[#+1&,start,MemberQ[list,#]&]);
%t A297838 a={1};b={2};c={3};n=0;
%t A297838 Do[{n++;
%t A297838   AppendTo[a,mex[Flatten[{a,b,c}],If[Length[a]==0,1,Last[a]]]],
%t A297838   AppendTo[b,mex[Flatten[{a,b,c}],Last[a]+n+1]],
%t A297838   AppendTo[c,Last[a]+Last[b]]},{z}];
%t A297838 Take[a,100] (* A297838 *)
%t A297838 Take[b,100] (* A298170 *)
%t A297838 Take[c,100] (* A298418 *)
%t A297838 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A297838 Cf. A299634, A298868, A297469, A298170, A298418.
%K A297838 nonn,easy
%O A297838 0,2
%A A297838 _Clark Kimberling_, Apr 25 2018