A297291 -id:A297291 - cp's OEIS frontend

A297292 Solution (b(n)) of the system of 3 complementary equations in Comments.

3, 6, 7, 11, 14, 15, 19, 20, 23, 25, 29, 30, 34, 36, 39, 40, 43, 46, 49, 50, 53, 55, 59, 60, 64, 65, 69, 70, 73, 75, 79, 80, 83, 86, 87, 91, 92, 95, 99, 100, 103, 106, 109, 110, 113, 115, 119, 120, 124, 126, 129, 130, 133, 135, 139, 140, 143, 146, 149, 150

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively:
a(n) = least new;
b(n) = least new > = a(n) + 2;
c(n) = a(n) + b(n) - 2;
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Conjectures:  for n >= 0,
0 <= 5*n + 4 - 2*a(n) <= 5,
0 <= 5*n + 8 - 2*b(n) <= 4,
0 <= c(n) - 5n <= 4.

			n:   0   1   2   3   4   5   6   7   8   9  10
a:   1   4   5   9  12  13  16  17  21  27  28
b:   3   6   7  11  14  15  19  20  23  25  29
c:   2   8  10  18  24  26  33  35  42  45  54

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A297291, A297293.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {};
Do[{AppendTo[a,
    mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
   AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
   AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
Take[a, 100]  (* A297291 *)
Take[b, 100]  (* A297292 *)
Take[c, 100]  (* A297293 *)
(* Peter J. C. Moses, Apr 23 2018 *)

A297293 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

2, 8, 10, 18, 24, 26, 33, 35, 42, 45, 54, 56, 63, 66, 74, 76, 82, 88, 94, 96, 102, 105, 114, 116, 123, 125, 134, 136, 142, 145, 154, 156, 162, 168, 170, 178, 180, 186, 194, 196, 202, 208, 214, 216, 222, 225, 234, 236, 243, 246, 254, 256, 262, 265, 274, 276

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively:
a(n) = least new;
b(n) = least new > = a(n) + 2;
c(n) = a(n) + b(n) - 2;
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Conjectures:  for n >= 0,
0 <= 5*n + 4 - 2*a(n) <= 5,
0 <= 5*n + 8 - 2*b(n) <= 4,
0 <= c(n) - 5n <= 4.

			n:   0   1   2   3   4   5   6   7   8   9  10
a:   1   4   5   9  12  13  16  17  21  27  28
b:   3   6   7  11  14  15  19  20  23  25  29
c:   2   8  10  18  24  26  33  35  42  45  54

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A297291, A297292.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {};
Do[{AppendTo[a,
    mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
   AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
   AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
Take[a, 100]  (* A297291 *)
Take[b, 100]  (* A297292 *)
Take[c, 100]  (* A297293 *)
(* Peter J. C. Moses, Apr 23 2018 *)

cp's OEIS Frontend

A297292 Solution (b(n)) of the system of 3 complementary equations in Comments.

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