A299405 Solution (a(n)) of the system of 5 complementary equations in Comments.

1, 5, 9, 14, 18, 22, 27, 31, 35, 39, 43, 48, 52, 56, 60, 65, 69, 73, 77, 82, 86, 90, 95, 99, 103, 107, 111, 116, 120, 124, 128, 133, 137, 141, 145, 150, 154, 158, 163, 167, 171, 175, 179, 184, 188, 192, 196, 201, 205, 209, 213, 218, 222, 226, 231, 235, 239

Offset: 0

Author

Clark Kimberling, Apr 22 2018

Keywords

• nonn
• easy

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).

Examples

			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
		

Links

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

Crossrefs

Cf. A036554, A299634, A299637, A299638, A299641, A299409.

Programs

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 *)