A356026 - OEIS

%I #16 Jan 18 2023 12:18:42

%S 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,

%T 20,14,32,58,78,76,52,38,68,74,59,67,101,27,47,88,75,61,109,50,124,54,

%U 113,41,102,119,84,34,40,136,105,71,92,131,108,28,171,169

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

%C This array appears in Guy, p. 360.

%C Conjectures involving a = A007063 and b = A356026:

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

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

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

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

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

%H Enrique Pérez Herrero, <a href="/A356026/b356026.txt">Table of n, a(n) for n = 1..10000</a>

%e Corner of the array (with terms of A356026 bracketed):

%e   [1]  2    3    4    5     6

%e    2  [3]   4    5    6     7

%e    2   4   [5]   6    7     8

%e    4   6    2   [7]   8     9

%e    2   8    6    9   [4]   10

%e    9  10    6   11    8   [12]

%t a = Join[{{1}},

%t    NestList[

%t     Flatten[{#, Range[Last[#] + 1, Last[#] + 3]} &[

%t        Flatten[Transpose[{Reverse[#[[1]]], #[[2]]} &[

%t           Partition[#, Length[#]/2] &[

%t            Drop[#, {(Length[#] + 1)/2}] &[#]]]]]]] &, {2, 3, 4}, 200]];

%t Take[a, 9] // TableForm;  (* the array, right-abbreviated *)

%t Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, a]] (* A356026 *)

%t (* _Peter J. C. Moses_, Jul 23 2022 *)

%t (* Alternate recursive code *)

%t KL[i_, j_] := i + j - 1 /; (j >= 2 i - 3);

%t KL[i_, j_] := KL[i - 1, i + (j - 2)/2] /; (EvenQ[j] && (j < 2 i - 3));

%t KL[i_, j_] := KL[i - 1, i - (j + 3)/2] /; (OddQ[j] && (j < 2 i - 3));

%t KL[i_] := KL[i] = KL[i, i]; SetAttributes[KL, Listable];

%t A356026[n_] := KL[n];

%t Array[A356026, 30]

%t (* _Enrique Pérez Herrero_, Jan 12 2023 *)

%o (PARI)

%o KL(i,j) =

%o {

%o my(i1,j1);

%o i1=i;

%o j1=j;

%o while(j1<(2*i1-3),

%o       if(j1%2,

%o          j1=i1-((j1+3)/2),

%o          j1=i1+((j1-2)/2)

%o        );

%o        i1--;

%o );

%o return(i1+j1-1);

%o }

%o A356026(i)=KL(i,i);

%o \\ _Enrique Pérez Herrero_, Jan 12 2023

%Y Cf. A007063, A355323.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jul 23 2022