A297465 - OEIS

%I #4 Apr 22 2018 17:50:23

%S 2,5,9,12,15,19,22,25,29,32,35,39,42,45,49,52,55,59,62,65,69,72,76,79,

%T 82,85,89,92,95,99,102,105,109,112,115,119,122,125,129,132,135,139,

%U 142,145,149,152,155,159,162,166,169,172,175,179,182,185,189,192

%N Solution (b(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 = A036554, the "anti-Fibonacci sequence."

%H Clark Kimberling, <a href="/A297465/b297465.txt">Table of n, a(n) for n = 0..1000</a>

%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, A297464, A297466,  A265389.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 22 2018