A296502 Solution (b(n)) of the system of 3 complementary equations in Comments.

2, 5, 14, 32, 53, 77, 104, 134, 170, 209, 254, 302, 353, 407, 464, 524, 587, 653, 722, 794, 869, 950, 1034, 1121, 1211, 1304, 1403, 1505, 1610, 1718, 1829, 1943, 2060, 2180, 2303, 2429, 2558, 2690, 2825, 2966, 3110, 3257, 3407, 3560, 3716, 3878, 4043, 4211

Offset

0,1

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A299634, A296484, A297149.

Sequence in context: A212393 A056358 A336229 * A036681 A125615 A096772

Adjacent sequences: A296499 A296500 A296501 * A296503 A296504 A296505

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2018

Status

approved