A299637 Solution (b(n)) of the system of 5 complementary equations in Comments.

2, 6, 11, 15, 19, 23, 28, 32, 36, 40, 44, 49, 53, 57, 61, 66, 70, 74, 79, 83, 87, 91, 96, 100, 104, 108, 112, 117, 121, 125, 129, 134, 138, 142, 147, 151, 155, 159, 164, 168, 172, 176, 181, 185, 189, 193, 197, 202, 206, 210, 215, 219, 223, 227, 232, 236, 240

Offset

0,1

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) = least new;

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

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

***

Conjecture: for all n >= 0,

0 <= 17n - 11 - 4 a(n) <= 4

0 <= 17n - 7 - 4 b(n) <= 4

0 <= 17n - 3 - 4 c(n) <= 3

0 <= 17n + 1 - 4 d(n) <= 3

0 <= 17n - 5 - e(n) <= 3

***

The sequences a,b,c,d,e partition the positive integers. The sequence e can be called the "anti-tetranacci sequence"; see A075326 (anti-Fibonacci numbers) and A265389 (anti-tribonacci numbers).

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  5   9   14  18  22  27  31  35  39
b:   2  6   11  15  19  23  28  32  36  40
c:   3  7   12  16  20  24  29  33  37  41
d:   4  8   13  17  21  25  30  34  38  42
e:  10  26  45  62  78  94 114 130 146 162

Mathematica

z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
Last[a] + Last[b] + Last[c] + Last[d]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
Take[a, 100]  (* A299405 *)
Take[b, 100]  (* A299637 *)
Take[c, 100]  (* A299638 *)
Take[d, 100]  (* A299641 *)
Take[e, 100]  (* A299409 *)

Crossrefs

Cf. A036554, A299634, A299405, A299638, A299641, A299409.

Sequence in context: A057244 A241672 A035487 * A190889 A020966 A193910

Adjacent sequences: A299634 A299635 A299636 * A299638 A299639 A299640

Keyword

nonn,easy

Author

Clark Kimberling, Apr 22 2018

Status

approved