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A299405 Solution (a(n)) of the system of 5 complementary equations in Comments. 6
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 (list; graph; refs; listen; history; text; internal format)
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) = 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, A299637, A299638,  A299641, A299409.

Sequence in context: A314826 A314827 A314828 * A314829 A287243 A314830

Adjacent sequences:  A299402 A299403 A299404 * A299406 A299407 A299408

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 22 2018

STATUS

approved

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Last modified January 17 15:12 EST 2020. Contains 330958 sequences. (Running on oeis4.)