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A298874 Solution (a(n)) of the system of 3 equations in Comments.

%I A298874 #7 Apr 21 2018 20:57:13
%S A298874 1,4,5,7,8,9,10,12,13,14,15,17,19,20,21,22,23,24,25,28,29,30,31,32,33,
%T A298874 34,36,37,38,39,40,41,42,44,46,47,48,49,50,51,52,53,54,55,56,58,59,61,
%U A298874 62,63,64,65,66,67,68,69,71,72,73,74,75,76,77,78,80,81
%N A298874 Solution (a(n)) of the system of 3 equations in Comments.
%C A298874 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298874 a(n) = least new;
%C A298874 b(n) = a(n) + b(n-1);
%C A298874 c(n) = a(n) + 2 b(n);
%C A298874 where "least new k" means the least positive integer not yet placed.
%C A298874 ***
%C A298874 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 A298874 Clark Kimberling, <a href="/A298874/b298874.txt">Table of n, a(n) for n = 0..1000</a>
%e A298874 n:   0    1    2    3    4    5    6    7    8    9
%e A298874 a:   1    4    5    7    8    9   10   12   13   14
%e A298874 b:   2    6   11   18   26   35   45   57   70   84
%e A298874 c:   3   16   27   43   60   30   79  100  126  153
%t A298874 z = 200;
%t A298874 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A298874 a = {1}; b = {2}; c = {3};
%t A298874 Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A298874    AppendTo[b, Last[a] + Last[b]],
%t A298874    AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
%t A298874 Take[a, 100]  (* A298874 *)
%t A298874 Take[b, 100]  (* A298875 *)
%t A298874 Take[c, 100]  (* A298876 *)
%Y A298874 Cf. A299634, A298871, A298875, A298876.
%K A298874 nonn,easy
%O A298874 0,2
%A A298874 _Clark Kimberling_, Apr 19 2018