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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 19 2018
STATUS
approved