A297464 - OEIS

%I #12 May 12 2018 21:30:17

%S 1,4,8,11,14,18,21,24,28,31,34,38,41,44,48,51,54,58,61,64,68,71,74,78,

%T 81,84,88,91,94,98,101,104,108,111,114,118,121,124,128,131,134,138,

%U 141,144,148,151,154,158,161,164,168,171,174,178,181,184,188,191

%N Solution (a(n)) of the system of 4 complementary equations in Comments.

%C Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:

%C a(n) = least new;

%C b(n) = least new;

%C c(n) = least new;

%C d(n) = a(n) + b(n) + c(n);

%C where "least new k" means the least positive integer not yet placed.

%C ***

%C Conjecture: for all n >= 0,

%C 0 <= 10n - 6 - 3 a(n) <= 2

%C 0 <= 10n - 2 - 3 b(n) <= 3

%C 0 <= 10n + 1 - 3 c(n) <= 3

%C 0 <= 10n - 3 - d(n) <= 2

%C ***

%C 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.

%H Clark Kimberling, <a href="/A297464/b297464.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = a(n-1) + a(n-3) - a(n-4) (conjectured).

%F d(n) = A275389(n) for n >= 0.

%e n:   0    1    2    3    4    5    6    7    8    9

%e a:   1    4    8   11   14   18   21   24   28   31

%e b:   2    5    9   12   15   19   22   25   29   32

%e c:   3    7   10   13   17   20   23   26   30   33

%e d:   6   16   27   36   46   57   66   75   87   96

%t z = 400;

%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];

%t Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],

%t    AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],

%t    AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],

%t    AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];

%t Take[a, 100]  (* A297464 *)

%t Take[b, 100]  (* A297465 *)

%t Take[c, 100]  (* A297466 *)

%t Take[d, 100]  (* A265389 *)

%Y Cf. A036554, A299634, A297465, A297466,  A265389.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Apr 19 2018