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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

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

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}];

{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 *)

CROSSREFS

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).

Sequence in context: A026183 A026199 A026207 * A261645 A193724 A266407

Adjacent sequences:  A309154 A309155 A309156 * A309158 A309159 A309160

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Jul 15 2019

STATUS

approved

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Last modified January 22 19:16 EST 2020. Contains 331153 sequences. (Running on oeis4.)