A299634 -id:A299634 - cp's OEIS frontend

A297469 Solution (bb(n)) of the system of 3 complementary equations in Comments.

2, 7, 11, 17, 22, 27, 31, 37, 41, 47, 51, 57, 62, 67, 71, 77, 82, 87, 91, 97, 102, 107, 111, 117, 121, 127, 131, 137, 142, 147, 151, 157, 161, 167, 171, 177, 182, 187, 191, 197, 201, 207, 211, 217, 222, 227, 231, 237, 242, 247, 251, 257, 262, 267, 271, 277

Offset: 0

Clark Kimberling, May 04 2018

Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:
aa(n) = least new;
bb(n) = aa(n) + cc(n-1);
cc(n) = least new;
where "least new k" means the least positive integer not yet placed.
***
The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >= 0,
(1) 5*n - 1 - 2*aa(n) is in {0,1,2},
(2) (aa(n) mod 5) is in {1,2,4},
(3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;
(4) (bb(n) mod 5) is in {1,2}.
From N. J. A. Sloane, Nov 05 2019: (Start)
Conjecture: For t >= 0, bb(2t) = 10t + 1 (+1 if binary expansion of t ends in an odd number of 0's), bb(2t+1) = 10t + 7.
The first part may also be written as bb(2t) = 10t + 1 + A328789(t-1).
(End)

			n:  0 1 2 3 4 5 6 7 8 9 10
aa: 1 4 6 9 12 14 16 19 21 24 26
bb: 2 7 11 17 22 27 31 37 41 47 51
cc: 3 5 8 10 13 15 18 20 23 25 28

Clark Kimberling, Table of n, a(n) for n = 0..10000 [This is the sequence bb]

Cf. A299634, A298468 (aa), A047218 (cc), A328789.

z = 500;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[b, Last[a] + Last[c]];
  AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
Take[a, 100] (* A298468 *)
Take[b, 100] (* A297469 *)
Take[c, 100] (* A047218 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - N. J. A. Sloane, Nov 03 2019

A298868 Solution (a(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 4, 6, 8, 11, 14, 15, 17, 19, 21, 24, 26, 27, 29, 32, 33, 34, 37, 41, 42, 45, 46, 48, 52, 53, 54, 57, 58, 59, 61, 64, 67, 70, 72, 73, 74, 77, 79, 82, 83, 87, 90, 92, 93, 94, 96, 98, 100, 101, 104, 105, 107, 111, 113, 115, 118, 119, 120, 122, 125, 126, 127

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, A298869, 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 *)

A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 4, 6, 9, 12, 14, 16, 19, 21, 24, 26, 29, 32, 34, 36, 39, 42, 44, 46, 49, 52, 54, 56, 59, 61, 64, 66, 69, 72, 74, 76, 79, 81, 84, 86, 89, 92, 94, 96, 99, 101, 104, 106, 109, 112, 114, 116, 119, 122, 124, 126, 129, 132, 134, 136, 139, 141, 144, 146, 149

Offset: 0

Clark Kimberling, May 04 2018

Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:
aa(n) = least new;
bb(n) = aa(n) + cc(n-1);
cc(n) = least new;
where "least new k" means the least positive integer not yet placed.
***
The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >=0,
(1) 5*n - 1 - 2*aa(n) is in {0,1,2},
(2) (aa(n) mod 5) is in {1,2,4},
(3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;
(4) (bb(n) mod 5) is in {1,2}.
From N. J. A. Sloane, Nov 05 2019: (Start)
Conjecture: For t >= 1, aa(2t) = 5t+1(+1 if binary expansion of t ends in an odd number of 0's), and for t >= 0, aa(2t+1) = 5t+4.
The first part may also be written as aa(2t) = 5t+1+A328789(t-1).
(End)

			n:  0 1 2 3 4 5 6 7 8 9 10
aa: 1 4 6 9 12 14 16 19 21 24 26
bb: 2 7 11 17 22 27 31 37 41 47 51
cc: 3 5 8 10 13 15 18 20 23 25 28

Clark Kimberling, Table of n, a(n) for n = 0..10000 [This is aa]

Cf. A299634, A297469 (bb), A047218 (cc), A328789.

z = 500;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[b, Last[a] + Last[c]];
  AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
Take[a, 100] (* A298468 *)
Take[b, 100] (* A297469 *)
Take[c, 100] (* A047218 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - N. J. A. Sloane, Nov 03 2019

A298871 Solution (a(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81

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, A298872, A298873, 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 *)

A298872 Solution (b(n)) of the system of 3 complementary 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 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, A298873, A298875.

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 *)

A299405 Solution (a(n)) of the system of 5 complementary equations in Comments.

Original entry on oeis.org

1, 5, 9, 14, 18, 22, 27, 31, 35, 39, 43, 48, 52, 56, 60, 65, 69, 73, 77, 82, 86, 90, 95, 99, 103, 107, 111, 116, 120, 124, 128, 133, 137, 141, 145, 150, 154, 158, 163, 167, 171, 175, 179, 184, 188, 192, 196, 201, 205, 209, 213, 218, 222, 226, 231, 235, 239

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, A299637, A299638, 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 *)

A299409 Solution (e(n)) of the system of 5 complementary equations in Comments.

Original entry on oeis.org

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

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
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.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 11, 18.

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

(* 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 *)

A297838 Solution (a(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

1, 4, 5, 7, 9, 12, 15, 16, 17, 20, 21, 25, 27, 28, 29, 33, 34, 35, 36, 39, 45, 46, 47, 48, 52, 56, 57, 58, 60, 61, 62, 64, 65, 67, 74, 75, 76, 78, 79, 80, 81, 87, 88, 94, 95, 97, 100, 102, 103, 104, 105, 106, 107, 108, 110, 114, 117, 123, 124, 125, 126, 127

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, A298170, 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 *)

A298874 Solution (a(n)) of the system of 3 equations in Comments.

Original entry on oeis.org

1, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81

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, A298875, 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 *)

A299637 Solution (b(n)) of the system of 5 complementary equations in Comments.

Original entry on oeis.org

2, 6, 11, 15, 19, 23, 28, 32, 36, 40, 44, 49, 53, 57, 61, 66, 70, 74, 79, 83, 87, 91, 96, 100, 104, 108, 112, 117, 121, 125, 129, 134, 138, 142, 147, 151, 155, 159, 164, 168, 172, 176, 181, 185, 189, 193, 197, 202, 206, 210, 215, 219, 223, 227, 232, 236, 240

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, A299638, 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 *)

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A297469 Solution (bb(n)) of the system of 3 complementary equations in Comments.

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A298868 Solution (a(n)) of the system of 3 complementary equations in Comments.

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A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments.

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A298871 Solution (a(n)) of the system of 3 complementary equations in Comments.

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A298872 Solution (b(n)) of the system of 3 complementary equations in Comments.

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A299405 Solution (a(n)) of the system of 5 complementary equations in Comments.

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A299409 Solution (e(n)) of the system of 5 complementary equations in Comments.

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A297838 Solution (a(n)) of the system of 3 complementary equations in Comments.

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