A297293 - OEIS

%I #10 May 01 2018 03:00:54

%S 2,8,10,18,24,26,33,35,42,45,54,56,63,66,74,76,82,88,94,96,102,105,

%T 114,116,123,125,134,136,142,145,154,156,162,168,170,178,180,186,194,

%U 196,202,208,214,216,222,225,234,236,243,246,254,256,262,265,274,276

%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) + 2;

%C c(n) = a(n) + b(n) - 2;

%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 Conjectures:  for n >= 0,

%C 0 <= 5*n + 4 - 2*a(n) <= 5,

%C 0 <= 5*n + 8 - 2*b(n) <= 4,

%C 0 <= c(n) - 5n <= 4.

%H Clark Kimberling, <a href="/A297293/b297293.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   4   5   9  12  13  16  17  21  27  28

%e b:   3   6   7  11  14  15  19  20  23  25  29

%e c:   2   8  10  18  24  26  33  35  42  45  54

%t z = 300;

%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t a = b = c = {};

%t Do[{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] + 2]],

%t    AppendTo[c, Last[a] + Last[b] - 2]}, {z}];

%t Take[a, 100]  (* A297291 *)

%t Take[b, 100]  (* A297292 *)

%t Take[c, 100]  (* A297293 *)

%t (* _Peter J. C. Moses_, Apr 23 2018 *)

%Y Cf. A299634, A297291, A297292.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 24 2018