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

1,2

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.

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

Example

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]

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 *)

Prog

(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

Crossrefs

Cf. A007063, A355323.

Sequence in context: A134487 A064537 A023899 * A356379 A324712 A279321

Adjacent sequences: A356023 A356024 A356025 * A356027 A356028 A356029

Keyword

nonn

Author

Clark Kimberling, Jul 23 2022

Status

approved