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A299409 Solution (e(n)) of the system of 5 complementary equations in Comments.

%I A299409 #21 Jun 23 2025 10:54:17
%S A299409 10,26,45,62,78,94,114,130,146,162,180,198,214,230,248,266,282,298,
%T A299409 317,334,350,366,386,402,418,434,451,470,486,502,520,538,554,570,589,
%U A299409 606,622,638,658,674,690,706,725,742,758,774,792,810,826,842,861,878,894
%N A299409 Solution (e(n)) of the system of 5 complementary equations in Comments.
%C A299409 Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
%C A299409 a(n) = least new;
%C A299409 b(n) = least new;
%C A299409 c(n) = least new;
%C A299409 d(n) = least new;
%C A299409 e(n) = a(n) + b(n) + c(n) + d(n);
%C A299409 where "least new k" means the least positive integer not yet placed.
%C A299409 ***
%C A299409 Conjecture: for all n >= 0,
%C A299409 0 <= 17n - 11 - 4 a(n) <= 4
%C A299409 0 <= 17n - 7 - 4 b(n) <= 4
%C A299409 0 <= 17n - 3 - 4 c(n) <= 3
%C A299409 0 <= 17n + 1 - 4 d(n) <= 3
%C A299409 0 <= 17n - 5 - e(n) <= 3
%C A299409 ***
%C A299409 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 A299409 Clark Kimberling, <a href="/A299409/b299409.txt">Table of n, a(n) for n = 0..1000</a>
%H A299409 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.
%H A299409 Robbert Fokkink and Gandhar Joshi, <a href="https://arxiv.org/abs/2506.13337">Anti-recurrence sequences</a>, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 11, 18.
%e A299409 n:   0  1   2    3   4   5   6   7   8   9
%e A299409 a:   1  5   9   14  18  22  27  31  35  39
%e A299409 b:   2  6   11  15  19  23  28  32  36  40
%e A299409 c:   3  7   12  16  20  24  29  33  37  41
%e A299409 d:   4  8   13  17  21  25  30  34  38  42
%e A299409 e:  10  26  45  62  78  94 114 130 146 162
%t A299409 (* Program 1: sequences a,b,c,d,e generated from the complementary equations *)
%t A299409 z = 200;
%t A299409 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299409 a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
%t A299409 Last[a] + Last[b] + Last[c] + Last[d]];
%t A299409 Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299409    AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299409    AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299409    AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
%t A299409    AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
%t A299409 Take[a, 100]  (* A299405 *)
%t A299409 Take[b, 100]  (* A299637 *)
%t A299409 Take[c, 100]  (* A299638 *)
%t A299409 Take[d, 100]  (* A299641 *)
%t A299409 Take[e, 100]  (* A299409 *)
%t A299409 (* Program 2: sequence e generated by iterating a morphism *)
%t A299409 morph = Nest[Flatten[# /. Thread[{0, 1, 2, 3} -> {{2, 3, 3, 1}, {2, 3, 2, 1}, {2, 3, 1, 1}, {2, 3, 0, 1}}]] &, {0}, 9];
%t A299409 A299409 = Accumulate[Prepend[Drop[Flatten[morph /. Thread[{0, 1, 2, 3} -> {{1, 1, 2, 4}, {1, 1, 3, 3}, {1, 1, 4, 2}, {1, 1, 5, 1}}]], 1] + 15, 10]];
%t A299409 Take[A299409, 100]  (* _Peter J. C. Moses_, May 04 2018 *)
%Y A299409 Cf. A036554, A299634, A299405, A299637,  A299638, A299641.
%K A299409 nonn,easy
%O A299409 0,1
%A A299409 _Clark Kimberling_, Apr 22 2018