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A299641 Solution (d(n)) of the system of 5 complementary equations in Comments.

%I A299641 #10 May 01 2018 03:00:33
%S A299641 4,8,13,17,21,25,30,34,38,42,47,51,55,59,64,68,72,76,81,85,89,93,98,
%T A299641 102,106,110,115,119,123,127,132,136,140,144,149,153,157,161,166,170,
%U A299641 174,178,183,187,191,195,200,204,208,212,217,221,225,229,234,238,242
%N A299641 Solution (d(n)) of the system of 5 complementary equations in Comments.
%C A299641 Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
%C A299641 a(n) = least new;
%C A299641 b(n) = least new;
%C A299641 c(n) = least new;
%C A299641 d(n) = least new;
%C A299641 e(n) = a(n) + b(n) + c(n) + d(n);
%C A299641 where "least new k" means the least positive integer not yet placed.
%C A299641 ***
%C A299641 Conjecture: for all n >= 0,
%C A299641 0 <= 17n - 11 - 4 a(n) <= 4
%C A299641 0 <= 17n - 7 - 4 b(n) <= 4
%C A299641 0 <= 17n - 3 - 4 c(n) <= 3
%C A299641 0 <= 17n + 1 - 4 d(n) <= 3
%C A299641 0 <= 17n - 5 - e(n) <= 3
%C A299641 ***
%C A299641 The sequences a,b,c,d,e partition the positive integers.  The sequence e can be called the "anti-tetranacci sequence"; see A075326 (anti-Fibonacci numbers) and A265389 (anti-tribonacci numbers).
%H A299641 Clark Kimberling, <a href="/A299641/b299641.txt">Table of n, a(n) for n = 0..1000</a>
%e A299641 n:   0  1   2    3   4   5   6   7   8   9
%e A299641 a:   1  5   9   14  18  22  27  31  35  39
%e A299641 b:   2  6   11  15  19  23  28  32  36  40
%e A299641 c:   3  7   12  16  20  24  29  33  37  41
%e A299641 d:   4  8   13  17  21  25  30  34  38  42
%e A299641 e:  10  26  45  62  78  94 114 130 146 162
%t A299641 z = 200;
%t A299641 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299641 a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
%t A299641 Last[a] + Last[b] + Last[c] + Last[d]];
%t A299641 Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299641    AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299641    AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299641    AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299641    AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
%t A299641 Take[a, 100]  (* A299405 *)
%t A299641 Take[b, 100]  (* A299637 *)
%t A299641 Take[c, 100]  (* A299638 *)
%t A299641 Take[d, 100]  (* A299641 *)
%t A299641 Take[e, 100]  (* A299409 *)
%Y A299641 Cf. A036554, A299634, A299405, A299637,  A299638, A299409.
%K A299641 nonn,easy
%O A299641 0,1
%A A299641 _Clark Kimberling_, Apr 22 2018