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A298170 Solution (b(n)) of the system of 3 complementary equations in Comments.

%I A298170 #17 Jul 27 2024 09:30:03
%S A298170 2,6,8,11,14,19,22,24,26,30,32,38,41,42,44,49,51,54,55,59,66,69,71,72,
%T A298170 77,83,84,86,90,92,93,96,99,101,109,112,113,116,119,121,122,130,131,
%U A298170 138,140,143,147,151,152,154,156,158,161,162,165,170,174,181,184
%N A298170 Solution (b(n)) of the system of 3 complementary equations in Comments.
%C A298170 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298170 a(n) = least new;
%C A298170 b(n) = least new > = a(n) + n + 1;
%C A298170 c(n) = a(n) + b(n);
%C A298170 where "least new k" means the least positive integer not yet placed.
%C A298170 ***
%C A298170 The sequences a,b,c partition the positive integers.
%C A298170 ***
%C A298170 Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C A298170 x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C A298170 x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
%C A298170 (The same limits occur in A298868 and A297469.)
%H A298170 Clark Kimberling, <a href="/A298170/b298170.txt">Table of n, a(n) for n = 0..1000</a>
%e A298170 n:   0   1   2   3   4    5   6   7   8   9  10
%e A298170 a:   1   4   5   7   9   12  15  16  17  20  21
%e A298170 b:   2   6   8  11   14  19  22  24  26  30  32
%e A298170 c:   3  10  13  18   23  31  37  40  43  50  53
%t A298170 z=200;
%t A298170 mex[list_,start_]:=(NestWhile[#+1&,start,MemberQ[list,#]&]);
%t A298170 a={1};b={2};c={3};n=0;
%t A298170 Do[{n++;
%t A298170   AppendTo[a,mex[Flatten[{a,b,c}],If[Length[a]==0,1,Last[a]]]],
%t A298170   AppendTo[b,mex[Flatten[{a,b,c}],Last[a]+n+1]],
%t A298170   AppendTo[c,Last[a]+Last[b]]},{z}];
%t A298170 Take[a,100] (* A297838 *)
%t A298170 Take[b,100] (* A298170 *)
%t A298170 Take[c,100] (* A298418 *)
%t A298170 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A298170 Cf. A299634, A298868, A297469, A297838, A298418.
%K A298170 nonn,easy
%O A298170 0,1
%A A298170 _Clark Kimberling_, Apr 25 2018