login
A299409
Solution (e(n)) of the system of 5 complementary equations in Comments.
6
10, 26, 45, 62, 78, 94, 114, 130, 146, 162, 180, 198, 214, 230, 248, 266, 282, 298, 317, 334, 350, 366, 386, 402, 418, 434, 451, 470, 486, 502, 520, 538, 554, 570, 589, 606, 622, 638, 658, 674, 690, 706, 725, 742, 758, 774, 792, 810, 826, 842, 861, 878, 894
OFFSET
0,1
COMMENTS
Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
a(n) = least new;
b(n) = least new;
c(n) = least new;
d(n) = least new;
e(n) = a(n) + b(n) + c(n) + d(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 17n - 11 - 4 a(n) <= 4
0 <= 17n - 7 - 4 b(n) <= 4
0 <= 17n - 3 - 4 c(n) <= 3
0 <= 17n + 1 - 4 d(n) <= 3
0 <= 17n - 5 - e(n) <= 3
***
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).
LINKS
EXAMPLE
n: 0 1 2 3 4 5 6 7 8 9
a: 1 5 9 14 18 22 27 31 35 39
b: 2 6 11 15 19 23 28 32 36 40
c: 3 7 12 16 20 24 29 33 37 41
d: 4 8 13 17 21 25 30 34 38 42
e: 10 26 45 62 78 94 114 130 146 162
MATHEMATICA
(* Program 1: sequences a, b, c, d, e generated from the complementary equations *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
Last[a] + Last[b] + Last[c] + Last[d]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
Take[a, 100] (* A299405 *)
Take[b, 100] (* A299637 *)
Take[c, 100] (* A299638 *)
Take[d, 100] (* A299641 *)
Take[e, 100] (* A299409 *)
(* Program 2: sequence e generated by iterating a morphism *)
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];
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]];
Take[A299409, 100] (* Peter J. C. Moses, May 04 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 22 2018
STATUS
approved