A299409 - OEIS

%I #14 May 05 2018 04:19:37

%S 10,26,45,62,78,94,114,130,146,162,180,198,214,230,248,266,282,298,

%T 317,334,350,366,386,402,418,434,451,470,486,502,520,538,554,570,589,

%U 606,622,638,658,674,690,706,725,742,758,774,792,810,826,842,861,878,894

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

%C Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:

%C a(n) = least new;

%C b(n) = least new;

%C c(n) = least new;

%C d(n) = least new;

%C e(n) = a(n) + b(n) + c(n) + d(n);

%C where "least new k" means the least positive integer not yet placed.

%C ***

%C Conjecture: for all n >= 0,

%C 0 <= 17n - 11 - 4 a(n) <= 4

%C 0 <= 17n - 7 - 4 b(n) <= 4

%C 0 <= 17n - 3 - 4 c(n) <= 3

%C 0 <= 17n + 1 - 4 d(n) <= 3

%C 0 <= 17n - 5 - e(n) <= 3

%C ***

%C 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 Clark Kimberling, <a href="/A299409/b299409.txt">Table of n, a(n) for n = 0..1000</a>

%e n:   0  1   2    3   4   5   6   7   8   9

%e a:   1  5   9   14  18  22  27  31  35  39

%e b:   2  6   11  15  19  23  28  32  36  40

%e c:   3  7   12  16  20  24  29  33  37  41

%e d:   4  8   13  17  21  25  30  34  38  42

%e e:  10  26  45  62  78  94 114 130 146 162

%t (* Program 1: sequences a,b,c,d,e generated from the complementary equations *)

%t z = 200;

%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,

%t Last[a] + Last[b] + Last[c] + Last[d]];

%t Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],

%t    AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],

%t    AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],

%t    AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],

%t    AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];

%t Take[a, 100]  (* A299405 *)

%t Take[b, 100]  (* A299637 *)

%t Take[c, 100]  (* A299638 *)

%t Take[d, 100]  (* A299641 *)

%t Take[e, 100]  (* A299409 *)

%t (* Program 2: sequence e generated by iterating a morphism *)

%t 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 = 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 Take[A299409, 100]  (* _Peter J. C. Moses_, May 04 2018 *)

%Y Cf. A036554, A299634, A299405, A299637,  A299638, A299641.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 22 2018