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A297466 Solution (c(n)) of the system of 4 complementary equations in Comments.

%I A297466 #14 Jul 17 2025 17:13:32
%S A297466 3,7,10,13,17,20,23,26,30,33,37,40,43,47,50,53,56,60,63,67,70,73,77,
%T A297466 80,83,86,90,93,97,100,103,107,110,113,116,120,123,127,130,133,137,
%U A297466 140,143,146,150,153,157,160,163,167,170,173,176,180,183,187,190,193
%N A297466 Solution (c(n)) of the system of 4 complementary equations in Comments.
%C A297466 Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
%C A297466 a(n) = least new;
%C A297466 b(n) = least new;
%C A297466 c(n) = least new;
%C A297466 d(n) = a(n) + b(n) + c(n);
%C A297466 where "least new k" means the least positive integer not yet placed.
%C A297466 ***
%C A297466 Conjecture: for all n >= 0,
%C A297466 0 <= 10n - 6 - 3 a(n) <= 2
%C A297466 0 <= 10n - 2 - 3 b(n) <= 3
%C A297466 0 <= 10n + 1 - 3 c(n) <= 3
%C A297466 0 <= 10n - 3 - d(n) <= 2
%C A297466 ***
%C A297466 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."
%C A297466 The linear recurrence with signature (1,0,0,0,0,0,0,0,1,-1) and g.f. (3 + 4*x + 3*x^2 + 3*x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8)/(1 - x - x^9 + x^10) first differs from this sequence at a(67). - _Georg Fischer_, Jul 17 2025
%H A297466 Clark Kimberling, <a href="/A297466/b297466.txt">Table of n, a(n) for n = 0..1000</a>
%e A297466 n:   0    1    2    3    4    5    6    7    8    9
%e A297466 a:   1    4    8   11   14   18   21   24   28   31
%e A297466 b:   2    5    9   12   15   19   22   25   29   32
%e A297466 c:   3    7   10   13   17   20   23   26   30   33
%e A297466 d:   6   16   27   36   46   57   66   75   87   96
%t A297466 z = 400;
%t A297466 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297466 a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
%t A297466 Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
%t A297466    AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
%t A297466    AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
%t A297466    AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
%t A297466 Take[a, 100]  (* A297464 *)
%t A297466 Take[b, 100]  (* A297465 *)
%t A297466 Take[c, 100]  (* A297466 *)
%t A297466 Take[d, 100]  (* A265389 *)
%Y A297466 Cf. A036554, A299634, A297464, A297465, A265389.
%K A297466 nonn,easy
%O A297466 0,1
%A A297466 _Clark Kimberling_, Apr 22 2018