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A326661 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(3n), where a(0) = 1; see Comments. 3
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 (list; table; 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 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).

LINKS

Table of n, a(n) for n=1..64.

EXAMPLE

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

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 *)

CROSSREFS

Cf. A309157, A326662, A326664.

Sequence in context: A257102 A226626 A249778 * A326662 A011049 A330883

Adjacent sequences:  A326658 A326659 A326660 * A326662 A326663 A326664

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jul 16 2019

STATUS

approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)