A299634 -id:A299634 - cp's OEIS frontend

A299638 Solution (c(n)) of the system of 5 complementary equations in Comments.

3, 7, 12, 16, 20, 24, 29, 33, 37, 41, 46, 50, 54, 58, 63, 67, 71, 75, 80, 84, 88, 92, 97, 101, 105, 109, 113, 118, 122, 126, 131, 135, 139, 143, 148, 152, 156, 160, 165, 169, 173, 177, 182, 186, 190, 194, 199, 203, 207, 211, 216, 220, 224, 228, 233, 237, 241

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) = 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).

			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

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

Cf. A036554, A299634, A299405, A299637, A299641, A299409.

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 *)

A299641 Solution (d(n)) of the system of 5 complementary equations in Comments.

Original entry on oeis.org

4, 8, 13, 17, 21, 25, 30, 34, 38, 42, 47, 51, 55, 59, 64, 68, 72, 76, 81, 85, 89, 93, 98, 102, 106, 110, 115, 119, 123, 127, 132, 136, 140, 144, 149, 153, 157, 161, 166, 170, 174, 178, 183, 187, 191, 195, 200, 204, 208, 212, 217, 221, 225, 229, 234, 238, 242

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) = 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).

			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

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

Cf. A036554, A299634, A299405, A299637, A299638, A299409.

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 *)

A297464 Solution (a(n)) of the system of 4 complementary equations in Comments.

Original entry on oeis.org

1, 4, 8, 11, 14, 18, 21, 24, 28, 31, 34, 38, 41, 44, 48, 51, 54, 58, 61, 64, 68, 71, 74, 78, 81, 84, 88, 91, 94, 98, 101, 104, 108, 111, 114, 118, 121, 124, 128, 131, 134, 138, 141, 144, 148, 151, 154, 158, 161, 164, 168, 171, 174, 178, 181, 184, 188, 191

Offset: 0

Clark Kimberling, Apr 19 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 = A075326, the "anti-Fibonacci sequence."  See A299409 for the "anti-tetranacci" sequences.

			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
Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025. See p. 8.

Cf. A036554, A299634, A297465, 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 *)

a(n) = a(n-1) + a(n-3) - a(n-4) (conjectured).

d(n) = A275389(n) for n >= 0.

A298873 Solution (c(n)) of the system of 3 complementary 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 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) + b(n-1);
c(n) = a(n) + 2 b(n);
where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.

			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, A298872, A298874.

z = 400;
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, mex[Flatten[{a, b, c}], Last[a] + Last[b]]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298871 *)
Take[b, 100]  (* A298872 *)
Take[c, 100]  (* A298873 *)

A298875 Solution (b(n)) of the system of 3 equations in Comments.

Original entry on oeis.org

2, 6, 11, 18, 26, 35, 45, 57, 70, 84, 99, 116, 135, 155, 176, 198, 221, 245, 270, 298, 327, 357, 388, 420, 453, 487, 523, 560, 598, 637, 677, 718, 760, 804, 850, 897, 945, 994, 1044, 1095, 1147, 1200, 1254, 1309, 1365, 1423, 1482, 1543, 1605, 1668, 1732

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.

			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, A298876.

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*)

A296484 Solution (a(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 3, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
a(n) = least new;
b(n) = a(n-1)+c(n-1);
c(n) = 2 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.

			n:   0   1    2    3    4    5    6    7   8     9
a:   1   3    6    7    8    9   10   12  13    15
b:   2   5   14   32   53   77  104  134  170  209
c:   4  11   26   46   69   95  124  158  196  239

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

Cf. A299634, A296502, A297149.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, a[[n - 1]] + c[[n - 1]]],
   AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100]  (* A296484 *)
Take[b, 100]  (* A296502 *)
Take[c, 100]  (* A297149 *)

A296502 Solution (b(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

2, 5, 14, 32, 53, 77, 104, 134, 170, 209, 254, 302, 353, 407, 464, 524, 587, 653, 722, 794, 869, 950, 1034, 1121, 1211, 1304, 1403, 1505, 1610, 1718, 1829, 1943, 2060, 2180, 2303, 2429, 2558, 2690, 2825, 2966, 3110, 3257, 3407, 3560, 3716, 3878, 4043, 4211

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
a(n) = least new;
b(n) = a(n-1)+c(n-1);
c(n) = 2 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.

			n:   0   1    2    3    4    5    6    7   8     9
a:   1   3    6    7    8    9   10   12  13    15
b:   2   5   14   32   53   77  104  134  170  209
c:   4  11   26   46   69   95  124  158  196  239

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

Cf. A299634, A296484, A297149.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, a[[n - 1]] + c[[n - 1]]],
   AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100]  (* A296484 *)
Take[b, 100]  (* A296502 *)
Take[c, 100]  (* A297149 *)

A297149 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

4, 11, 26, 46, 69, 95, 124, 158, 196, 239, 286, 336, 389, 445, 504, 566, 631, 699, 770, 844, 923, 1006, 1092, 1181, 1273, 1370, 1471, 1575, 1682, 1792, 1905, 2021, 2140, 2262, 2387, 2515, 2646, 2780, 2919, 3062, 3208, 3357, 3509, 3664, 3824, 3988, 4155, 4325

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
a(n) = least new;
b(n) = a(n-1)+c(n-1);
c(n) = 2 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.

			n:   0   1    2    3    4    5    6    7   8     9
a:   1   3    6    7    8    9   10   12  13    15
b:   2   5   14   32   53   77  104  134  170  209
c:   4  11   26   46   69   95  124  158  196  239

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

Cf. A299634, A296484, A296502.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, a[[n - 1]] + c[[n - 1]]],
   AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100]  (* A296484 *)
Take[b, 100]  (* A296502 *)
Take[c, 100]  (* A297149 *)

A297291 Solution (a(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 4, 5, 9, 12, 13, 16, 17, 21, 22, 27, 28, 31, 32, 37, 38, 41, 44, 47, 48, 51, 52, 57, 58, 61, 62, 67, 68, 71, 72, 77, 78, 81, 84, 85, 89, 90, 93, 97, 98, 101, 104, 107, 108, 111, 112, 117, 118, 121, 122, 127, 128, 131, 132, 137, 138, 141, 144, 147, 148

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, A297292, 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 *)

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A299638 Solution (c(n)) of the system of 5 complementary equations in Comments.

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A299641 Solution (d(n)) of the system of 5 complementary equations in Comments.

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A297464 Solution (a(n)) of the system of 4 complementary equations in Comments.

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A298873 Solution (c(n)) of the system of 3 complementary equations in Comments.

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A298875 Solution (b(n)) of the system of 3 equations in Comments.

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A296484 Solution (a(n)) of the system of 3 complementary equations in Comments.

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A296502 Solution (b(n)) of the system of 3 complementary equations in Comments.

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A297149 Solution (c(n)) of the system of 3 complementary equations in Comments.

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