A299423 - OEIS

%I #8 May 05 2018 04:18:11

%S 4,7,16,21,24,29,32,37,44,49,56,63,66,71,78,83,88,91,98,103,106,113,

%T 116,121,128,131,136,143,147,152,154,164,168,173,180,185,189,191,200,

%U 203,210,214,219,225,234,237,240,243,250,255,262,267,272,275,281,291

%N Solution (c(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="/A299423/b299423.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, A299533.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, May 01 2018