A297149 Solution (c(n)) of the system of 3 complementary equations in Comments.

4, 11, 26, 46, 69, 95, 124, 158, 196, 239, 286, 336, 389, 445, 504, 566, 631, 699, 770, 844, 923, 1006, 1092, 1181, 1273, 1370, 1471, 1575, 1682, 1792, 1905, 2021, 2140, 2262, 2387, 2515, 2646, 2780, 2919, 3062, 3208, 3357, 3509, 3664, 3824, 3988, 4155, 4325

Offset: 0

Author

Clark Kimberling, Apr 24 2018

Keywords

• nonn
• easy

Comments

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:

a(n) = least new;

b(n) = a(n-1)+c(n-1);

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

where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.

Examples

			n:   0   1    2    3    4    5    6    7   8     9
a:   1   3    6    7    8    9   10   12  13    15
b:   2   5   14   32   53   77  104  134  170  209
c:   4  11   26   46   69   95  124  158  196  239
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A299634, A296484, A296502.

Programs

Mathematica

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, a[[n - 1]] + c[[n - 1]]],
   AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100]  (* A296484 *)
Take[b, 100]  (* A296502 *)
Take[c, 100]  (* A297149 *)