A298873 Solution (c(n)) of the system of 3 complementary equations in Comments.

3, 16, 27, 43, 60, 79, 100, 126, 153, 182, 213, 249, 289, 330, 373, 418, 465, 514, 565, 624, 683, 744, 807, 872, 939, 1008, 1082, 1157, 1234, 1313, 1394, 1477, 1562, 1652, 1746, 1841, 1938, 2037, 2138, 2241, 2346, 2453, 2562, 2673, 2786, 2904, 3023, 3147

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, A298872, A298874.

Sequence in context: A335308 A193367 A322413 * A298876 A258717 A013196

Adjacent sequences: A298870 A298871 A298872 * A298874 A298875 A298876

Keyword

nonn,easy

Author

Clark Kimberling, Apr 18 2018

Status

approved