A297464 Solution (a(n)) of the system of 4 complementary equations in Comments.

1, 4, 8, 11, 14, 18, 21, 24, 28, 31, 34, 38, 41, 44, 48, 51, 54, 58, 61, 64, 68, 71, 74, 78, 81, 84, 88, 91, 94, 98, 101, 104, 108, 111, 114, 118, 121, 124, 128, 131, 134, 138, 141, 144, 148, 151, 154, 158, 161, 164, 168, 171, 174, 178, 181, 184, 188, 191

Offset

0,2

Comments

Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:

a(n) = least new;

b(n) = least new;

c(n) = least new;

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

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

***

Conjecture: for all n >= 0,

0 <= 10n - 6 - 3 a(n) <= 2

0 <= 10n - 2 - 3 b(n) <= 3

0 <= 10n + 1 - 3 c(n) <= 3

0 <= 10n - 3 - d(n) <= 2

***

The sequences a,b,c,d partition the positive integers. The sequence d can be called the "anti-tribonacci sequence"; viz., if sequences a and b are defined as above, and c(n) is defined by c(n) = a(n) + b(n), then the resulting system of 3 complementary sequences gives c = A075326, the "anti-Fibonacci sequence." See A299409 for the "anti-tetranacci" sequences.

Links

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

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) (conjectured).

d(n) = A275389(n) for n >= 0.

Example

n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    8   11   14   18   21   24   28   31
b:   2    5    9   12   15   19   22   25   29   32
c:   3    7   10   13   17   20   23   26   30   33
d:   6   16   27   36   46   57   66   75   87   96

Mathematica

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
Take[a, 100]  (* A297464 *)
Take[b, 100]  (* A297465 *)
Take[c, 100]  (* A297466 *)
Take[d, 100]  (* A265389 *)

Crossrefs

Cf. A036554, A299634, A297465, A297466, A265389.

Sequence in context: A113553 A342279 A248228 * A311034 A311035 A311036

Adjacent sequences: A297461 A297462 A297463 * A297465 A297466 A297467

Keyword

nonn,easy

Author

Clark Kimberling, Apr 19 2018

Status

approved