cp's OEIS Frontend

A297465 Solution (b(n)) of the system of 4 complementary equations in Comments.

%I A297465 #4 Apr 22 2018 17:50:23
%S A297465 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 A297465 82,85,89,92,95,99,102,105,109,112,115,119,122,125,129,132,135,139,
%U A297465 142,145,149,152,155,159,162,166,169,172,175,179,182,185,189,192
%N A297465 Solution (b(n)) of the system of 4 complementary equations in Comments.
%C A297465 Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:
%C A297465 a(n) = least new;
%C A297465 b(n) = least new;
%C A297465 c(n) = least new;
%C A297465 d(n) = a(n) + b(n) + c(n);
%C A297465 where "least new k" means the least positive integer not yet placed.
%C A297465 ***
%C A297465 Conjecture: for all n >= 0,
%C A297465 0 <= 10n - 6 - 3 a(n) <= 2
%C A297465 0 <= 10n - 2 - 3 b(n) <= 3
%C A297465 0 <= 10n +1 - 3 c(n) <= 3
%C A297465 0 <= 10n - 3 - d(n) <= 2
%C A297465 ***
%C A297465 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 A297465 Clark Kimberling, <a href="/A297465/b297465.txt">Table of n, a(n) for n = 0..1000</a>
%e A297465 n:   0    1    2    3    4    5    6    7    8    9
%e A297465 a:   1    4    8   11   14   18   21   24   28   31
%e A297465 b:   2    5    9   12   15   19   22   25   29   32
%e A297465 c:   3    7   10   13   17   20   23   26   30   33
%e A297465 d:   6   16   27   36   46   57   66   75   87   96
%t A297465 z = 400;
%t A297465 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297465 a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
%t A297465 Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
%t A297465    AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
%t A297465    AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
%t A297465    AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
%t A297465 Take[a, 100]  (* A297464 *)
%t A297465 Take[b, 100]  (* A297465 *)
%t A297465 Take[c, 100]  (* A297466 *)
%t A297465 Take[d, 100]  (* A265389 *)
%Y A297465 Cf. A036554, A299634, A297464, A297466,  A265389.
%K A297465 nonn,easy
%O A297465 0,1
%A A297465 _Clark Kimberling_, Apr 22 2018