login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A297464 Solution (a(n)) of the system of 4 complementary equations in Comments. 4
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 (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) = 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 05:50 EST 2022. Contains 358578 sequences. (Running on oeis4.)