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).

EXAMPLE

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

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 *)

CROSSREFS

KEYWORD

AUTHOR

Clark Kimberling, Jul 16 2019

EXTENSIONS

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

STATUS

approved