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Solution (b(n)) of the system of 3 equations in Comments.
4

%I #4 Apr 21 2018 20:57:20

%S 2,6,11,18,26,35,45,57,70,84,99,116,135,155,176,198,221,245,270,298,

%T 327,357,388,420,453,487,523,560,598,637,677,718,760,804,850,897,945,

%U 994,1044,1095,1147,1200,1254,1309,1365,1423,1482,1543,1605,1668,1732

%N Solution (b(n)) of the system of 3 equations in Comments.

%C Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:

%C a(n) = least new;

%C b(n) = a(n) + b(n-1);

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

%C where "least new k" means the least positive integer not yet placed.

%C ***

%C Do these sequences a,b,c partition the positive integers? They differ from the corresponding partitioning sequences A298871, A298872, and A298872. For example, A298872(56) = 2139, whereas A298875(56) = 2138.

%H Clark Kimberling, <a href="/A298875/b298875.txt">Table of n, a(n) for n = 0..1000</a>

%e n: 0 1 2 3 4 5 6 7 8 9

%e a: 1 4 5 7 8 9 10 12 13 14

%e b: 2 6 11 18 26 35 45 57 70 84

%e c: 3 16 27 43 60 30 79 100 126 153

%t z = 200;

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

%t a = {1}; b = {2}; c = {3};

%t Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],

%t AppendTo[b, Last[a] + Last[b]],

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

%t Take[a, 100] (*A298874*)

%t Take[b, 100] (*A298875*)

%t Take[c, 100] (*A298876*)

%Y Cf. A299634, A298871, A298874, A298876.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 19 2018