A298872 Solution (b(n)) of the system of 3 complementary equations in Comments.

2, 6, 11, 18, 26, 35, 45, 57, 70, 84, 99, 116, 135, 155, 176, 198, 221, 245, 270, 298, 327, 357, 388, 420, 453, 487, 523, 560, 598, 637, 677, 718, 760, 804, 850, 897, 945, 994, 1044, 1095, 1147, 1200, 1254, 1309, 1365, 1423, 1482, 1543, 1605, 1668, 1732

Offset

0,1

Comments

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

a(n) = least new;

b(n) = least new k >= a(n) + b(n-1);

c(n) = a(n) + 2 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    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 = 400;
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, mex[Flatten[{a, b, c}], Last[a] + Last[b]]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298871 *)
Take[b, 100]  (* A298872 *)
Take[c, 100]  (* A298873 *)

Crossrefs

Cf. A299634, A298871, A298873, A298875.

Sequence in context: A104813 A239698 A039745 * A298875 A329598 A248469

Adjacent sequences: A298869 A298870 A298871 * A298873 A298874 A298875

Keyword

nonn,easy

Author

Clark Kimberling, Apr 18 2018

Status

approved