A326662 -id:A326662 - cp's OEIS frontend

A309231 Column 3 of the array at A326662 see Comments.

7, 17, 25, 34, 43, 53, 61, 71, 79, 89, 97, 106, 115, 125, 133, 142, 151, 161, 169, 178, 187, 197, 205, 215, 223, 233, 241, 250, 259, 269, 277, 287, 295, 305, 313, 322, 331, 341, 349, 359, 367, 377, 385, 394, 403, 413, 421, 430, 439, 449, 457, 466, 475, 485

Offset: 1

Clark Kimberling, Jul 16 2019

This is the sequence (c(n)) defined by the complementary equation c(n) = a(2n) + b(2n), with initial value a(1) = 1. See A326662. Conjecture: 9n-c(n) is in {1,2} for all n.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A326662.

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}]; c
(* Peter J. C. Moses, Jul 04 2019 *)

A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 5, 3, 4, 12, 6, 7, 20, 8, 9, 26, 10, 11, 33, 13, 14, 41, 15, 16, 47, 17, 18, 54, 19, 21, 61, 22, 23, 68, 24, 25, 75, 27, 28, 83, 29, 30, 89, 31, 32, 96, 34, 35, 104, 36, 37, 110, 38, 39, 117, 40, 42, 124, 43, 44, 131, 45, 46, 138, 48, 49, 146, 50, 51

Offset: 1

Clark Kimberling, Jul 15 2019

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for these 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).

			c(1) = a(1) + b(2) > = 1 + 3, so that
a(2) = mex{1,2} = 3;
b(2) = mex{1,2,3} = 4;
c(1) = 5.
Then c(2) = a(2) + b(4) >= 3 + 8, so that
a(3) = 6, b(3) = 7;
a(4) = 8, b(4) = 9;
c(2) = a(2) + b(4) = 3 + 9 = 12.
   n    a(n) b(n) c(n)
  --------------------
   1      1    2    5
   2      3    4   12
   3      6    7   20
   4      8    9   26
   5     10   11   33
   6     13   14   41
   7     15   16   47
   8     17   18   54
   9     19   21   61
  10     22   23   68

Clark Kimberling and Peter J. C. Moses, Complementary Equations with Advanced Subscripts, J. Int. Seq. 24 (2021) Article 21.3.3.

Cf. A326663 (3rd column),
A101544 solves c(n) = a(n) + b(n),
A326661 solves c(n) = a(n) + b(3n),
A326662 solves c(n) = a(2n) + b(2n).

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}];
{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 *)

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

Clark Kimberling, Jul 16 2019

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).

			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

Cf. A309157, A326662, A326664.

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 *)

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

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A309231 Column 3 of the array at A326662 see Comments.

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A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

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

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Mathematica

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