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 1-3 of 3 results.

A326663 Column 3 of the array at A309157; see Comments.

Original entry on oeis.org

5, 12, 20, 26, 33, 41, 47, 54, 61, 68, 75, 83, 89, 96, 104, 110, 117, 124, 131, 138, 146, 152, 159, 167, 173, 180, 188, 194, 201, 209, 215, 222, 230, 236, 243, 250, 257, 264, 272, 278, 285, 293, 299, 306, 313, 320, 327, 335, 341, 348, 356, 362, 369, 377, 383
Offset: 1

Views

Author

Clark Kimberling, Jul 16 2019

Keywords

Comments

This is the sequence (c(n)) defined by the complementary equation c(n) = a(n) + b(2n), with initial value a(1) = 1. See A309157. Conjecture: 7n-c(n) is in {1,2} for all n.

Links

Crossrefs

Cf. A309157.

Programs

  • Mathematica
    mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 1; k = 2;
Do[Do[AppendTo[a,
  mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}]; c
(* Peter J. C. Moses, Jul 04 2019 *)

A326661 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(3n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 7, 3, 4, 16, 5, 6, 25, 8, 9, 35, 10, 11, 43, 12, 13, 52, 14, 15, 61, 17, 18, 71, 19, 20, 79, 21, 22, 88, 23, 24, 97, 26, 27, 107, 28, 29, 115, 30, 31, 124, 32, 33, 133, 34, 36, 142, 37, 38, 151, 39, 40, 160, 41, 42, 169, 44, 45, 179, 46, 47, 187, 48
Offset: 1

Views

Author

Clark Kimberling, Jul 16 2019

Keywords

Comments

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for the following conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).

Examples

			c(1) = a(1) + b(3) >= 1 + 6, so that b(1) = mex{1} = 2; a(2) = mex{1,2} = 3; b(2) = mex{1,2,3} = 4; a(3)= mex{1,2,3,4} = 5, a(4) = mex{1,2,3,4,5} = 6, c(1) = 7.
n           a(n)      b(n)     c(n)
---------------------------
1             1        2        7
2             3        4       16
3             5        6       25
4             8        9       35
5            10       11       43
6            12       13       52
7            14       15       61
8            17       18       74
9            19       20       79
10           21       22       88

Crossrefs

Cf. A309157, A326662, A326664.

Programs

  • Mathematica
    mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 1; k = 3;
Do[Do[AppendTo[a,
  mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
{a, b, c} // ColumnForm
a = Take[a, Length[c]]; b = Take[b, Length[c]];
Flatten[Transpose[{a, b, c}]](* Peter J. C. Moses, Jul 04 2019 *)

Extensions

Replaced a(0)->a(1) in NAME. - R. J. Mathar, Jun 19 2021

A326662 Rectangular array in 3 columns that solve the complementary equation c(n) = a(2n) + b(2n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 7, 3, 4, 17, 5, 6, 25, 8, 9, 34, 10, 11, 43, 12, 13, 53, 14, 15, 61, 16, 18, 71, 19, 20, 79, 21, 22, 89, 23, 24, 97, 26, 27, 106, 28, 29, 115, 30, 31, 125, 32, 33, 133, 35, 36, 142, 37, 38, 151, 39, 40, 161, 41, 42, 169, 44, 45, 178, 46, 47, 187, 48
Offset: 1

Views

Author

Clark Kimberling, Jul 16 2019

Keywords

Comments

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for the following conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).

Examples

			c(1) = a(2) + b(2) >= 3 + 4, so that b(1) = mex{1} = 2; a(2) = mex{1,2} = 3; b(2) = mex{1,2,3} = 4; a(3)= mex{1,2,3,4} = 5, a(4) = mex{1,2,3,4,5} = 6, c(1) = 7.
n           a(n)      b(n)     c(n)
-----------------------------------
1             1        2        7
2             3        4       17
3             5        6       25
4             8        9       34
5            10       11       43
6            12       13       53
7            14       15       61
8            16       18       71
9            19       20       79
10           21       22       89

Crossrefs

Cf. A309157, A326661.

Programs

  • Mathematica
    mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 2; k = 2;
Do[Do[AppendTo[a,
   mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
{a, b, c} // ColumnForm
a = Take[a, Length[c]]; b = Take[b, Length[c]];
Flatten[Transpose[{a, b, c}]](* Peter J. C. Moses, Jul 04 2019 *)

Extensions

Replaced a(0)->a(1) in NAME. - R. J. Mathar, Jun 19 2021
Showing 1-3 of 3 results.

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