A299641 Solution (d(n)) of the system of 5 complementary equations in Comments.

4, 8, 13, 17, 21, 25, 30, 34, 38, 42, 47, 51, 55, 59, 64, 68, 72, 76, 81, 85, 89, 93, 98, 102, 106, 110, 115, 119, 123, 127, 132, 136, 140, 144, 149, 153, 157, 161, 166, 170, 174, 178, 183, 187, 191, 195, 200, 204, 208, 212, 217, 221, 225, 229, 234, 238, 242

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, A299637, A299638, A299409.

Sequence in context: A311816 A081843 A004938 * A311817 A311818 A311819

Adjacent sequences: A299638 A299639 A299640 * A299642 A299643 A299644

Keyword

nonn,easy

Author

Clark Kimberling, Apr 22 2018

Status

approved