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.

Previous Showing 31-34 of 34 results.

A297467 Solution (a(n)) of the system of 2 complementary equations in Comments.

Original entry on oeis.org

1, 2, 10, 31, 35, 95, 99, 108, 112, 289, 293, 302, 306, 330, 335, 343, 348, 875, 880, 888, 893, 916, 921, 929, 934, 1002, 1007, 1018, 1023, 1043, 1048, 1059, 1064, 2641, 2646, 2657, 2662, 2682, 2687, 2698, 2703, 2768, 2773, 2784, 2789, 2809, 2814, 2825, 2830
Offset: 0

Views

Author

Clark Kimberling, Apr 24 2018

Keywords

Comments

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, a(1) = 1, b(0) = 3; for n >= 1,
a(2n) = 3*a(n) + b(n);
a(2n+1) = 3*a(n-1) + n;
b(n) = least new;
where "least new k" means the least positive integer not yet placed. The sequences (a(n)) and (b(n)) are complementary.

Examples

			n:   0  1   2   3   4   5   6   7   8
a:   1  2  10  31  35  95  99 108 112
b:   3  4   5   6   7   8   9  11  12

Links

Crossrefs

Cf. A299634, A297468.

Programs

  • Mathematica
    z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1, 2}; b = {3};
Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]];
AppendTo[a, 3 a[[#/2 + 1]] + b[[#/2 + 1]]] &[Length[a]];
AppendTo[a, 3 a[[(# + 3)/2]] + (# - 1)/2] &[Length[a]], {z}]
Take[a, 100]  (* A297467 *)
Take[b, 100]  (* A297468 *)
(* Peter J. C. Moses, Apr 22 2018 *)

A297468 Solution (b(n)) of the system of 2 complementary equations in Comments.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
Offset: 0

Views

Author

Clark Kimberling, Apr 24 2018

Keywords

Comments

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, a(1) = 1, b(0) = 3; for n >= 1,
a(2n) = 3*a(n) + b(n);
a(2n+1) = 3*a(n-1) + n;
b(n) = least new;
where "least new k" means the least positive integer not yet placed. The sequences (a(n)) and (b(n)) are complementary.

Examples

			n:   0  1   2   3   4   5   6   7   8
a:   1  2  10  31  35  95  99 108 112
b:   3  4   5   6   7   8   9  11  12

Links

Crossrefs

Cf. A299634, A297467.

Programs

  • Mathematica
    z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1, 2}; b = {3};
Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]];
 AppendTo[a, 3 a[[#/2 + 1]] + b[[#/2 + 1]]] &[Length[a]];
 AppendTo[a, 3 a[[(# + 3)/2]] + (# - 1)/2] &[Length[a]], {z}]
Take[a, 100]  (* A297467 *)
Take[b, 100]  (* A297468 *)
(* Peter J. C. Moses,  Apr 22 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 *)
Previous Showing 31-34 of 34 results.

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