A299634 -id:A299634 - cp's OEIS frontend

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

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 *)

A297465 Solution (b(n)) of the system of 4 complementary equations in Comments.

Original entry on oeis.org

2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 35, 39, 42, 45, 49, 52, 55, 59, 62, 65, 69, 72, 76, 79, 82, 85, 89, 92, 95, 99, 102, 105, 109, 112, 115, 119, 122, 125, 129, 132, 135, 139, 142, 145, 149, 152, 155, 159, 162, 166, 169, 172, 175, 179, 182, 185, 189, 192

Offset: 0

Clark Kimberling, Apr 22 2018

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) = a(n) + b(n) + c(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 10n - 6 - 3 a(n) <= 2
0 <= 10n - 2 - 3 b(n) <= 3
0 <= 10n +1 - 3 c(n) <= 3
0 <= 10n - 3 - d(n) <= 2
***
The sequences a,b,c,d partition the positive integers.  The sequence d can be called the "anti-tribonacci sequence"; viz., if sequences a and b are defined as above, and c(n) is defined by c(n) = a(n) + b(n), then the resulting system of 3 complementary sequences gives c = A036554, the "anti-Fibonacci sequence."

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    8   11   14   18   21   24   28   31
b:   2    5    9   12   15   19   22   25   29   32
c:   3    7   10   13   17   20   23   26   30   33
d:   6   16   27   36   46   57   66   75   87   96

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

Cf. A036554, A299634, A297464, A297466, A265389.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
Take[a, 100]  (* A297464 *)
Take[b, 100]  (* A297465 *)
Take[c, 100]  (* A297466 *)
Take[d, 100]  (* A265389 *)

A297466 Solution (c(n)) of the system of 4 complementary equations in Comments.

Original entry on oeis.org

3, 7, 10, 13, 17, 20, 23, 26, 30, 33, 37, 40, 43, 47, 50, 53, 56, 60, 63, 67, 70, 73, 77, 80, 83, 86, 90, 93, 97, 100, 103, 107, 110, 113, 116, 120, 123, 127, 130, 133, 137, 140, 143, 146, 150, 153, 157, 160, 163, 167, 170, 173, 176, 180, 183, 187, 190, 193

Offset: 0

Clark Kimberling, Apr 22 2018

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) = a(n) + b(n) + c(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 10n - 6 - 3 a(n) <= 2
0 <= 10n - 2 - 3 b(n) <= 3
0 <= 10n + 1 - 3 c(n) <= 3
0 <= 10n - 3 - d(n) <= 2
***
The sequences a,b,c,d partition the positive integers.  The sequence d can be called the "anti-tribonacci sequence"; viz., if sequences a and b are defined as above, and c(n) is defined by c(n) = a(n) + b(n), then the resulting system of 3 complementary sequences gives c = A036554, the "anti-Fibonacci sequence."
The linear recurrence with signature (1,0,0,0,0,0,0,0,1,-1) and g.f. (3 + 4*x + 3*x^2 + 3*x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8)/(1 - x - x^9 + x^10) first differs from this sequence at a(67). - Georg Fischer, Jul 17 2025

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    8   11   14   18   21   24   28   31
b:   2    5    9   12   15   19   22   25   29   32
c:   3    7   10   13   17   20   23   26   30   33
d:   6   16   27   36   46   57   66   75   87   96

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

Cf. A036554, A299634, A297464, A297465, A265389.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {}; AppendTo[d, Last[a] + Last[b] + Last[c]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d}], 1]],
   AppendTo[d, Last[a] + Last[b] + Last[c]]}, {z}];
Take[a, 100]  (* A297464 *)
Take[b, 100]  (* A297465 *)
Take[c, 100]  (* A297466 *)
Take[d, 100]  (* A265389 *)

A298170 Solution (b(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

2, 6, 8, 11, 14, 19, 22, 24, 26, 30, 32, 38, 41, 42, 44, 49, 51, 54, 55, 59, 66, 69, 71, 72, 77, 83, 84, 86, 90, 92, 93, 96, 99, 101, 109, 112, 113, 116, 119, 121, 122, 130, 131, 138, 140, 143, 147, 151, 152, 154, 156, 158, 161, 162, 165, 170, 174, 181, 184

Offset: 0

Clark Kimberling, Apr 25 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = least new > = a(n) + n + 1;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
(The same limits occur in A298868 and A297469.)

			n:   0   1   2   3   4    5   6   7   8   9  10
a:   1   4   5   7   9   12  15  16  17  20  21
b:   2   6   8  11   14  19  22  24  26  30  32
c:   3  10  13  18   23  31  37  40  43  50  53

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

Cf. A299634, A298868, A297469, A297838, A298418.

z=200;
mex[list_,start_]:=(NestWhile[#+1&,start,MemberQ[list,#]&]);
a={1};b={2};c={3};n=0;
Do[{n++;
  AppendTo[a,mex[Flatten[{a,b,c}],If[Length[a]==0,1,Last[a]]]],
  AppendTo[b,mex[Flatten[{a,b,c}],Last[a]+n+1]],
  AppendTo[c,Last[a]+Last[b]]},{z}];
Take[a,100] (* A297838 *)
Take[b,100] (* A298170 *)
Take[c,100] (* A298418 *)
(* Peter J. C. Moses, Apr 23 2018 *)

A298418 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

3, 10, 13, 18, 23, 31, 37, 40, 43, 50, 53, 63, 68, 70, 73, 82, 85, 89, 91, 98, 111, 115, 118, 120, 129, 139, 141, 144, 150, 153, 155, 160, 164, 168, 183, 187, 189, 194, 198, 201, 203, 217, 219, 232, 235, 240, 247, 253, 255, 258, 261, 264, 268, 270, 275, 284

Offset: 0

Clark Kimberling, May 01 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = least new > = a(n) + n + 1;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
(The same limits occur in A298868 and A297469.)

			n:   0   1   2   3   4    5   6   7   8   9  10
a:   1   4   5   7   9   12  15  16  17  20  21
b:   2   6   8  11   14  19  22  24  26  30  32
c:   3  10  13  18   23  31  37  40  43  50  53

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

Cf. A299634, A298868, A297469, A297838, A298170.

z=200;
mex[list_,start_]:=(NestWhile[#+1&,start,MemberQ[list,#]&]);
a={1};b={2};c={3};n=0;
Do[{n++;
  AppendTo[a,mex[Flatten[{a,b,c}],If[Length[a]==0,1,Last[a]]]],
  AppendTo[b,mex[Flatten[{a,b,c}],Last[a]+n+1]],
  AppendTo[c,Last[a]+Last[b]]},{z}];
Take[a,100] (* A297838 *)
Take[b,100] (* A298170 *)
Take[c,100] (* A298418 *)
(* Peter J. C. Moses, Apr 23 2018 *)

A298869 Solution (b(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

2, 5, 7, 10, 12, 16, 20, 22, 25, 28, 31, 36, 38, 40, 43, 47, 50, 51, 56, 60, 63, 66, 68, 71, 76, 78, 81, 85, 86, 89, 91, 95, 99, 103, 106, 109, 110, 114, 117, 121, 124, 128, 133, 135, 137, 139, 142, 146, 148, 151, 154, 156, 159, 164, 167, 170, 174, 176, 178

Offset: 0

Clark Kimberling, Apr 17 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = least new k >= a(n) + n;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers. Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67));
x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n -> 2x+1.

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    6    8   11   14   15   17   19   21
b:   2    5    7   10   12   16   20   22   25   28
c:   3    9   13   18   23   30   35   39   44   49

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

Cf. A299634, A298868, A298870.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],
   AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A298868 *)
Take[b, 100] (* A298869 *)
Take[c, 100] (* A298870 *)
(* Peter J. C. Moses, Apr 08 2018 *)

A298870 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

3, 9, 13, 18, 23, 30, 35, 39, 44, 49, 55, 62, 65, 69, 75, 80, 84, 88, 97, 102, 108, 112, 116, 123, 129, 132, 138, 143, 145, 150, 155, 162, 169, 175, 179, 183, 187, 193, 199, 204, 211, 218, 225, 228, 231, 235, 240, 246, 249, 255, 259, 263, 270, 277, 282, 288

Offset: 0

Clark Kimberling, Apr 18 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = least new k >= a(n) + n;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers. Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    6    8   11   14   15   17   19   21
b:   2    5    7   10   12   16   20   22   25   28
c:   3    9   13   18   23   30   35   39   44   49

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

Cf. A299634, A298868, A298869.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],
   AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A298868 *)
Take[b, 100] (* A298869 *)
Take[c, 100] (* A298870 *)
(* Peter J. C. Moses, Apr 08 2018 *)

A298876 Solution (c(n)) of the system of 3 equations in Comments.

Original entry on oeis.org

3, 16, 27, 43, 60, 79, 100, 126, 153, 182, 213, 249, 289, 330, 373, 418, 465, 514, 565, 624, 683, 744, 807, 872, 939, 1008, 1082, 1157, 1234, 1313, 1394, 1477, 1562, 1652, 1746, 1841, 1938, 2037, 2138, 2241, 2346, 2453, 2562, 2673, 2786, 2904, 3023, 3147

Offset: 0

Clark Kimberling, Apr 19 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = a(n) + b(n-1);
c(n) = a(n) + 2 b(n);
where "least new k" means the least positive integer not yet placed.
***
Do these sequences a,b,c partition the positive integers?  They differ from the corresponding partitioning sequences A298871, A298872, and A298872.  For example, A298872(56) = 2139, whereas A298875(56) = 2138.
Differs from A298873 first at n=56. - Georg Fischer, Oct 10 2018

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    5    7    8    9   10   12   13   14
b:   2    6   11   18   26   35   45   57   70   84
c:   3   16   27   43   60   30   79  100  126  153

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

Cf. A299634, A298871, A298874, A298875.

z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, Last[a] + Last[b]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298874 *)
Take[b, 100]  (* A298875 *)
Take[c, 100]  (* A298876 *)

A299635 Solution (b(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

2, 5, 6, 7, 8, 11, 13, 15, 18, 20, 21, 24, 25, 27, 29, 31, 32, 33, 34, 35, 38, 39, 43, 44, 45, 46, 47, 51, 52, 53, 55, 56, 59, 60, 63, 65, 69, 71, 72, 73, 75, 79, 80, 81, 82, 84, 85, 87, 91, 93, 96, 98, 99, 100, 105, 107, 109, 111, 113, 114, 116, 117, 119

Offset: 0

Clark Kimberling, Apr 17 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new k >= 2*b(n-1);
b(n) = least new k;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Let x = 11/6.  Conjectures:
a(n) - 2*n*x = 0 for infinitely many n;
b(n) - n*x = 0 for infinitely many n;
c(n) - 3*n*x = 0 for infinitely many n;
(a(n) - 2*n*x) is unbounded below and above;
(b(n) - n*x) is unbounded below and above;
(c(n) - 3*n*x) is unbounded below and above;
***
Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
2 occurs infinitely many times in d(c).

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4   10   12   14   17   23   26   30   37
b:   2    5    6    7    8   11   13   15   18   20
c:   3    9   16   19   22   28   36   41   48   57

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

Cf. A299634, A299636.

z = 1000;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
Do[{
  AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
  AppendTo[b, mex[Flatten[{a, b, c}], 1]],
  AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A299634 *)
Take[b, 100] (* A299635 *)
Take[c, 100] (* A299636 *)
(* Peter J. C. Moses, Apr 08 2018 *)

A299636 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

3, 9, 16, 19, 22, 28, 36, 41, 48, 57, 61, 66, 74, 77, 83, 89, 94, 97, 101, 103, 108, 115, 121, 130, 133, 136, 139, 146, 154, 157, 161, 166, 171, 178, 183, 191, 200, 209, 214, 217, 222, 229, 238, 241, 244, 248, 253, 257, 265, 275, 282, 290, 295, 298, 306, 317

Offset: 0

Clark Kimberling, Apr 17 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new k >= 2*b(n-1);
b(n) = least new k;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Let x = 11/6.  Conjectures:
a(n) - 2*n*x = 0 for infinitely many n;
b(n) - n*x = 0 for infinitely many n;
c(n) - 3*n*x = 0 for infinitely many n;
(a(n) - 2*n*x) is unbounded below and above;
(b(n) - n*x) is unbounded below and above;
(c(n) - 3*n*x) is unbounded below and above;
***
Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
2 occurs infinitely many times in d(c).

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4   10   12   14   17   23   26   30   37
b:   2    5    6    7    8   11   13   15   18   20
c:   3    9   16   19   22   28   36   41   48   57

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

Cf. A299634, A299635.

z = 1000;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
Do[{
  AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
  AppendTo[b, mex[Flatten[{a, b, c}], 1]],
  AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A299634 *)
Take[b, 100] (* A299635 *)
Take[c, 100] (* A299636 *)
(* Peter J. C. Moses, Apr 08 2018 *)

cp's OEIS Frontend

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

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A297465 Solution (b(n)) of the system of 4 complementary equations in Comments.

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A297466 Solution (c(n)) of the system of 4 complementary equations in Comments.

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A298170 Solution (b(n)) of the system of 3 complementary equations in Comments.

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A298418 Solution (c(n)) of the system of 3 complementary equations in Comments.

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A298869 Solution (b(n)) of the system of 3 complementary equations in Comments.

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A298870 Solution (c(n)) of the system of 3 complementary equations in Comments.

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A298876 Solution (c(n)) of the system of 3 equations in Comments.

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