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A299637 Solution (b(n)) of the system of 5 complementary equations in Comments.

%I A299637 #10 May 01 2018 03:00:28
%S A299637 2,6,11,15,19,23,28,32,36,40,44,49,53,57,61,66,70,74,79,83,87,91,96,
%T A299637 100,104,108,112,117,121,125,129,134,138,142,147,151,155,159,164,168,
%U A299637 172,176,181,185,189,193,197,202,206,210,215,219,223,227,232,236,240
%N A299637 Solution (b(n)) of the system of 5 complementary equations in Comments.
%C A299637 Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
%C A299637 a(n) = least new;
%C A299637 b(n) = least new;
%C A299637 c(n) = least new;
%C A299637 d(n) = least new;
%C A299637 e(n) = a(n) + b(n) + c(n) + d(n);
%C A299637 where "least new k" means the least positive integer not yet placed.
%C A299637 ***
%C A299637 Conjecture: for all n >= 0,
%C A299637 0 <= 17n - 11 - 4 a(n) <= 4
%C A299637 0 <= 17n - 7 - 4 b(n) <= 4
%C A299637 0 <= 17n - 3 - 4 c(n) <= 3
%C A299637 0 <= 17n + 1 - 4 d(n) <= 3
%C A299637 0 <= 17n - 5 - e(n) <= 3
%C A299637 ***
%C A299637 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 A299637 Clark Kimberling, <a href="/A299637/b299637.txt">Table of n, a(n) for n = 0..1000</a>
%e A299637 n:   0  1   2    3   4   5   6   7   8   9
%e A299637 a:   1  5   9   14  18  22  27  31  35  39
%e A299637 b:   2  6   11  15  19  23  28  32  36  40
%e A299637 c:   3  7   12  16  20  24  29  33  37  41
%e A299637 d:   4  8   13  17  21  25  30  34  38  42
%e A299637 e:  10  26  45  62  78  94 114 130 146 162
%t A299637 z = 200;
%t A299637 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299637 a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
%t A299637 Last[a] + Last[b] + Last[c] + Last[d]];
%t A299637 Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299637    AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299637    AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299637    AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299637    AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
%t A299637 Take[a, 100]  (* A299405 *)
%t A299637 Take[b, 100]  (* A299637 *)
%t A299637 Take[c, 100]  (* A299638 *)
%t A299637 Take[d, 100]  (* A299641 *)
%t A299637 Take[e, 100]  (* A299409 *)
%Y A299637 Cf. A036554, A299634, A299405, A299638,  A299641, A299409.
%K A299637 nonn,easy
%O A299637 0,1
%A A299637 _Clark Kimberling_, Apr 22 2018