A356026 Main diagonal of right-and-left variant of Kimberling expulsion array, A007063.

1, 3, 5, 7, 4, 12, 10, 17, 6, 22, 15, 19, 24, 33, 31, 18, 8, 44, 35, 9, 39, 55, 26, 42, 29, 20, 14, 32, 58, 78, 76, 52, 38, 68, 74, 59, 67, 101, 27, 47, 88, 75, 61, 109, 50, 124, 54, 113, 41, 102, 119, 84, 34, 40, 136, 105, 71, 92, 131, 108, 28, 171, 169

Offset: 1

Author

Clark Kimberling, Jul 23 2022

Keywords

• nonn

Comments

This array appears in Guy, p. 360.

Conjectures involving a = A007063 and b = A356026:

(1) Every positive integer is eventually expelled in a and in b.

(2) a(n) < b(n) for infinitely many n.

(3) a(n) > b(n) for infinitely many n.

(4) a(n) = b(n) for infinitely many n; see A355323.

Examples

			Corner of the array (with terms of A356026 bracketed):
  [1]  2    3    4    5     6
   2  [3]   4    5    6     7
   2   4   [5]   6    7     8
   4   6    2   [7]   8     9
   2   8    6    9   [4]   10
   9  10    6   11    8   [12]
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004; Section E35.

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007063, A355323.

Programs

Mathematica

a = Join[{{1}},
   NestList[
    Flatten[{#, Range[Last[#] + 1, Last[#] + 3]} &[
       Flatten[Transpose[{Reverse[#[[1]]], #[[2]]} &[
          Partition[#, Length[#]/2] &[
           Drop[#, {(Length[#] + 1)/2}] &[#]]]]]]] &, {2, 3, 4}, 200]];
Take[a, 9] // TableForm;  (* the array, right-abbreviated *)
Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, a]] (* A356026 *)
(* Peter J. C. Moses, Jul 23 2022 *)
(* Alternate recursive code *)
KL[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
KL[i_, j_] := KL[i - 1, i + (j - 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
KL[i_, j_] := KL[i - 1, i - (j + 3)/2] /; (OddQ[j] && (j < 2 i - 3));
KL[i_] := KL[i] = KL[i, i]; SetAttributes[KL, Listable];
A356026[n_] := KL[n];
Array[A356026, 30]
(* Enrique Pérez Herrero, Jan 12 2023 *)

PARI

KL(i,j) =
{
my(i1,j1);
i1=i;
j1=j;
while(j1<(2*i1-3),
      if(j1%2,
         j1=i1-((j1+3)/2),
         j1=i1+((j1-2)/2)
       );
       i1--;
);
return(i1+j1-1);
}
A356026(i)=KL(i,i);
\\ Enrique Pérez Herrero, Jan 12 2023