cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Clark Kimberling, Apr 25 2018

Keywords

Comments

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.)

Examples

			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

Links

Crossrefs

Cf. A299634, A298868, A297469, A298170, A298418.

Programs

  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Apr 25 2018

Keywords

Comments

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.)

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, May 01 2018

Keywords

Comments

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.)

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Apr 17 2018

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A299634, A298868, A298870.

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Apr 18 2018

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A299634, A298868, A298869.

Programs

  • Mathematica
    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 *)

A299423 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

4, 7, 16, 21, 24, 29, 32, 37, 44, 49, 56, 63, 66, 71, 78, 83, 88, 91, 98, 103, 106, 113, 116, 121, 128, 131, 136, 143, 147, 152, 154, 164, 168, 173, 180, 185, 189, 191, 200, 203, 210, 214, 219, 225, 234, 237, 240, 243, 250, 255, 262, 267, 272, 275, 281, 291
Offset: 0

Views

Author

Clark Kimberling, May 01 2018

Keywords

Comments

Define sequences a(n), b(n), c(n) recursively:
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 A297838.)

Examples

			n:   0   1   2   3   4   5   6   7   8   9  10
a:   1   2   6   8   9  11  12  14  17  19  22
b:   3   5  10  13  15  18  20  23  27  30  34
c:   4   7  16  21  24  29  32  37  44  49  56

Links

Crossrefs

Cf. A299634, A298868, A297838, A297469, A299533.

Programs

  • Mathematica
    z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {}; 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}];
(* Peter J. C. Moses, Apr 23 2018 *)
Take[a, 100] (* A297469 *)
Take[b, 100] (* A299533 *)
Take[c, 100] (* A299423 *)
(* Peter J. C. Moses, Apr 23 2018 *)

A299533 Solution (b(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

3, 5, 10, 13, 15, 18, 20, 23, 27, 30, 34, 38, 40, 43, 47, 50, 53, 55, 59, 62, 64, 68, 70, 73, 77, 79, 82, 86, 89, 92, 93, 99, 101, 104, 108, 111, 114, 115, 120, 122, 126, 129, 132, 135, 140, 142, 144, 146, 150, 153, 157, 160, 163, 165, 169, 174, 176, 178
Offset: 0

Views

Author

Clark Kimberling, May 01 2018

Keywords

Comments

Define sequences a(n), b(n), c(n) recursively:
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 A297838.)

Examples

			n:   0   1   2   3   4   5   6   7   8   9  10
a:   1   2   6   8   9  11  12  14  17  19  22
b:   3   5  10  13  15  18  20  23  27  30  34
c:   4   7  16  21  24  29  32  37  44  49  56

Links

Crossrefs

Cf. A299634, A298868, A297838, A297469, A299423.

Programs

  • Mathematica
    z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {}; 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}];
(* Peter J. C. Moses, Apr 23 2018 *)
Take[a, 100] (* A297469 *)
Take[b, 100] (* A299533 *)
Take[c, 100] (* A299423 *)
(* Peter J. C. Moses, Apr 23 2018 *)
Showing 1-7 of 7 results.

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