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

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, 34, 36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81

Offset

0,2

Comments

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

a(n) = least new;

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

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

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

***

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.

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    4    5    7    8    9   10   12   13   14
b:   2    6   11   18   26   35   45   57   70   84
c:   3   16   27   43   60   30   79  100  126  153

Mathematica

z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, Last[a] + Last[b]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298874 *)
Take[b, 100]  (* A298875 *)
Take[c, 100]  (* A298876 *)

Crossrefs

Cf. A299634, A298871, A298875, A298876.

Sequence in context: A182757 A164515 A298871 * A266267 A189477 A047494

Adjacent sequences: A298871 A298872 A298873 * A298875 A298876 A298877

Keyword

nonn,easy

Author

Clark Kimberling, Apr 19 2018

Status

approved