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A297464 Solution (a(n)) of the system of 4 complementary equations in Comments.

%I A297464 #15 Mar 07 2025 09:06:49
%S A297464 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 A297464 81,84,88,91,94,98,101,104,108,111,114,118,121,124,128,131,134,138,
%U A297464 141,144,148,151,154,158,161,164,168,171,174,178,181,184,188,191
%N A297464 Solution (a(n)) of the system of 4 complementary equations in Comments.
%C A297464 Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:
%C A297464 a(n) = least new;
%C A297464 b(n) = least new;
%C A297464 c(n) = least new;
%C A297464 d(n) = a(n) + b(n) + c(n);
%C A297464 where "least new k" means the least positive integer not yet placed.
%C A297464 ***
%C A297464 Conjecture: for all n >= 0,
%C A297464 0 <= 10n - 6 - 3 a(n) <= 2
%C A297464 0 <= 10n - 2 - 3 b(n) <= 3
%C A297464 0 <= 10n + 1 - 3 c(n) <= 3
%C A297464 0 <= 10n - 3 - d(n) <= 2
%C A297464 ***
%C A297464 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 A297464 Clark Kimberling, <a href="/A297464/b297464.txt">Table of n, a(n) for n = 0..1000</a>
%H A297464 Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, <a href="https://arxiv.org/abs/2503.04122">Using Walnut to solve problems from the OEIS</a>, arXiv:2503.04122 [math.NT], 2025. See p. 8.
%F A297464 a(n) = a(n-1) + a(n-3) - a(n-4) (conjectured).
%F A297464 d(n) = A275389(n) for n >= 0.
%e A297464 n:   0    1    2    3    4    5    6    7    8    9
%e A297464 a:   1    4    8   11   14   18   21   24   28   31
%e A297464 b:   2    5    9   12   15   19   22   25   29   32
%e A297464 c:   3    7   10   13   17   20   23   26   30   33
%e A297464 d:   6   16   27   36   46   57   66   75   87   96
%t A297464 z = 400;
%t A297464 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297464 a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
%t A297464 Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
%t A297464    AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
%t A297464    AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
%t A297464    AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
%t A297464 Take[a, 100]  (* A297464 *)
%t A297464 Take[b, 100]  (* A297465 *)
%t A297464 Take[c, 100]  (* A297466 *)
%t A297464 Take[d, 100]  (* A265389 *)
%Y A297464 Cf. A036554, A299634, A297465, A297466,  A265389.
%K A297464 nonn,easy
%O A297464 0,2
%A A297464 _Clark Kimberling_, Apr 19 2018