

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
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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 "antitribonacci 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 "antiFibonacci sequence." See A299409 for the "antitetranacci" sequences.


LINKS

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


FORMULA

a(n) = a(n1) + a(n3)  a(n4) (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



