A035486 -id:A035486 - cp's OEIS frontend

A006852 Step at which n is expelled in Kimberling's puzzle (A035486).

1, 25, 2, 4, 3, 22, 6, 8, 10, 5, 32, 83, 44, 14, 7, 66, 169, 11, 49595, 9, 69, 16, 24, 12, 43, 47, 7598, 15, 133, 109, 13, 198, 19, 33, 18, 23, 58, 65, 60, 93167, 68, 17, 1523, 39, 75, 20, 99, 34, 117, 123

Offset: 1

N. J. A. Sloane

R. K. Guy, Unsolved Problems Number Theory, Sect E35.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Enrique Pérez Herrero [1..11000], Goudout Élie [11001..20000], Table of n, a(n) for n = 1..20000
D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998. [From _Enrique Pérez Herrero_, Mar 28 2010]
C. Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991.

Cf. A007063.

Cf. A175312. - Enrique Pérez Herrero, Mar 28 2010

L[n_] := L[n] = (
i = Floor[(n + 4)/3];
j = Floor[(2*n + 1)/3];
While[(i != j), j = Max[2*(i - j), 2*(j - i) - 1]; i++ ];
Return[i];
) A006852[n_] := L[n]
(* Enrique Pérez Herrero, Mar 28 2010 *)

A006852(n)=
{
my(i,j);
i=floor((n+4)/3);
j=floor((2*n+1)/3);
while((i!=j),
j=max(2*i-2*j,-1-2*i+2*j);
i++;
); return(i); }
\\ Enrique Pérez Herrero, Feb 25 2010

a(n) >= floor((n+4)/3), n is expulsed from the unshuffled zone. - Enrique Pérez Herrero, Feb 25 2010

7593 corrected to 7598 by Hans Havermann, July 1998

A007063 Main diagonal of Kimberling's expulsion array (A035486).

Original entry on oeis.org

1, 3, 5, 4, 10, 7, 15, 8, 20, 9, 18, 24, 31, 14, 28, 22, 42, 35, 33, 46, 53, 6, 36, 23, 2, 55, 62, 59, 76, 65, 54, 11, 34, 48, 70, 79, 99, 95, 44, 97, 58, 84, 25, 13, 122, 83, 26, 115, 82, 91, 52, 138, 67, 90, 71, 119, 64, 37, 81, 39, 169, 88, 108, 141, 38, 16, 146, 41, 21

Offset: 1

N. J. A. Sloane, Mira Bernstein

From Clark Kimberling Aug 05 2022, Oct 24 2022: (Start)
Eight such arrays (including A035486 and A356026) have been coded by Peter J. C. Moses using
R for "right side of expelled (number)",
L for "left side",
I for "inner", i.e., next to expelled, and
O for "outer", i.e., farthest from expelled. For example, the array A035486 (and diagonal A007063) are coded as RILI. For the eight codes see Example and Mathematica. It is conjectured that six of the eight diagonal sequences are permutations of the positive integers. (End)

			The eight diagonals described in Comments:
A007063 = RILI = (1, 3, 5, 4, 10,  7, 15,  8, 20,  9, 18, 24, 31, 14, ... )
A282348 = ROLO = (1, 3, 5, 2,  8,  9,  4, 10,  7, 20, 12, 24, 14, 23, ... )
A356376 = LORO = (1, 3, 5, 6,  4, 11, 12,  9, 13, 15, 23,  7, 27, 16, ... )
A356026 = LIRI = (1, 3, 5, 7,  4, 12, 10, 17,  6, 22, 15, 19, 24, 33, ... )
A356377 = ROLI = (1, 3, 5, 4,  8,  6, 10, 15,  2,  9, 13, 26, 11, 12, ... )
A356378 = RILO = (1, 3, 5, 2, 10,  9, 15,  8, 20, 19,  7, 21, 31,  6, ... )
A356379 = LORI = (1, 3, 5, 7,  4, 12, 11, 17, 10, 22, 21,  9, 23, 33, ... )
A356380 = LIRO = (1, 3, 5, 6,  4, 11, 13,  2,  7, 14, 24,  9, 10, 31, ... )

D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.
R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004; Section E35, p. 359.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Enrique Pérez Herrero, Table of n, a(n) for n = 1..100000
Enrique Pérez Herrero, Kimberling's Expulsion Array
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991; Solution to Problem 1615, Crux Mathematicorum, Vol. 18, March 1992, pp. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence

Cf. A175312, A006852, A035486, A038807, A282348, A356376, A356026, A356377, A356378, A356379, A356380.

K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - j/2 - 1] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
A007063[i_] := A007063[i] = K[i, i]; SetAttributes[A007063, Listable] (* Enrique Pérez Herrero, Feb 09 2010 *)
(* Next program generates the 8 arrays with highlighted diagonal sequences. *)
len = 1000;
roli = Join[{{1}},
   NestList[
    Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1],
         Range[(Length[#] - 1)/2, 1, -1]]]],
      Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
rili = Join[{{1}},
  NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
        Range[(Length[#] - 1)/2, 1, -1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
   len]];(*A007063*)
rolo = Join[{{1}},
  NestList[Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1],
        Range[1, (Length[#] - 1)/2]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
   len]];(*A282348*)
rilo = Join[{{1}},
  NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
        Range[1, (Length[#] - 1)/2, 1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
lori = Join[{{1}},
   NestList[
    Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
         Range[(Length[#] + 3)/2, Length[#]]]]],
      Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
liri = Join[{{1}},
  NestList[Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
        Range[(Length[#] + 3)/2, Length[#]]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4},
   len]];(*A356026*)
loro = Join[{{1}},
  NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
        Range[Length[#], (Length[#] + 3)/2, -1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
liro = Join[{{1}},
   NestList[
    Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
         Range[Length[#], (Length[#] + 3)/2, -1]]]],
      Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
(Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 200] &[
      ToExpression[#]]} &, {"rolo", "rilo", "roli", "rili", "loro",
    "liro", "lori", "liri"}]) // ColumnForm
rows = 10; Map[{#,
   Grid[Map[Map[StringPadLeft[ToString[#], 2] &, #] &,
     Take[ToExpression[#], rows]],
    Frame -> {None, None, Map[{#, #} -> True &, Range[rows]]},
    FrameStyle -> Directive[Red]]} &, {"rolo", "rilo", "roli", "rili",
   "loro", "liro", "lori", "liri"}]
(* Peter J. C. Moses, Oct 24 2022; Clark Kimberling, Oct 24 2022 *)

K(i,j) = { my(i1,j1);i1=i; j1=j;
while(j1<(2*i1-3),if(j1%2,j1=i1+((j1-1)/2),j1=i1-((j1+2)/2));i1--;);
return(i1+j1-1);}
A007063(i)=K(i,i); \\ Enrique Pérez Herrero, Feb 21 2010

a(theta(k)) = 3*theta(k)-(k+1), where theta(k) = Sum_{i=0..k-1} 2^floor(i/3). - Enrique Pérez Herrero, Feb 23 2010
From Connor Brown, May 05 2023 to Feb 01 2024: (Start)
14 sets of values which predictably appear within the sequence have been found, 1 by Richard Guy (1992) and 13 by Connor Brown (2023). Below, k is any positive integer unless otherwise specified.
a(3*2^k-3) = 9*2^k -  3*k - 10. (Guy, 1992)
a(5*2^k-3) = 15*2^k - 3*k - 12.
a(4*2^k-3) = 12*2^k - 3*k - 11.
a((20/3)*2^k-(4/3)) = 20*2^k - 3*k - 13 for odd k.
a((16/3)*2^k-(4/3)) = 16*2^k - 3*k - 12 for even k.
a((40/7)*2^k-(3/7)) = (120/7)*2^k - 3*k - (93/7) for k==1 (mod 3).
a((16/5)*2^k-(2/5)) = (48/5)*2^k - 3*k - (46/5) for k==1 (mod 4).
a((12/5)*2^k-(2/5)) = (36/5)*2^k - 3*k - (41/5) for k==0 (mod 4).
a((48/13)*2^k+(8/13)) = (144/13)*2^k - 3*k - (145/13) for k==1 (mod 12).
a((64/13)*2^k+(8/13)) = (192/13)*2^k - 3*k - (158/13) for k==3 (mod 12).
a((80/13)*2^k+(8/13)) = (240/13)*2^k - 3*k - (171/13) for k==8 (mod 12).
a((64/9)*2^k+(7/9)) = (64/3)*2^k - 3*k - (35/3) for k==1 (mod 6).
a((80/9)*2^k+(7/9)) = (80/3)*2^k - 3*k - (38/3) for k==4 (mod 6).
a((64/15)*2^k+(7/15)) = (64/5)*2^k - 3*k - (63/5) for k>4, k==1 (mod 4).
(End)
a(n) <= A175312(n). - Enrique Pérez Herrero, Dec 14 2024

More terms from James Sellers, Dec 23 1999

A038807 Future of the smallest-perizeroin komet in Kimberling's expulsion array (A035486).

Original entry on oeis.org

2, 3, 5, 10, 9, 20, 46, 83, 12, 24, 23, 36, 79, 124, 172, 56, 119, 61, 169, 17, 42, 84, 232, 285, 596, 1186, 3190, 6857, 14225, 12495, 30482, 45827, 79090, 144112, 423486, 1087497, 2443796, 628733, 871389, 1199242, 2787410, 7975876

Offset: 0

Hans Havermann

nonn

Could the komet be a planit?

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.
Hans Havermann, Algorithm, #4, 1992, p. 2.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..74
Lars Blomberg & Hans Havermann, komets & planits (250 kometary path fragments)
Hans Havermann, A Recreational Endeavour
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991; Solution to Problem 1615, Crux Mathematicorum, Vol. 18, March 1992, pp. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence

Cf. A007063, A006852, A038834.

K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
A007063[i_] := K[i];
A038807[1] := 2;
A038807[n_] := A007063[A038807[n - 1]];
ReleaseHold[Table[A038807[n], {n, 1, 35}]]
(* Enrique Pérez Herrero, Jan 11 2023 *)

a(0) = 2; a(n) = a(n-1)-th term in Kimberling's expulsion array (A007063).

A175312 Maximum value on Lower Shuffle Part of Kimberling's Expulsion Array (A035486).

Original entry on oeis.org

1, 3, 5, 7, 10, 12, 15, 17, 20, 22, 25, 28, 31, 33, 36, 39, 42, 44, 47, 50, 53, 55, 58, 61, 64, 67, 70, 73, 76, 78, 81, 84, 87, 90, 93, 96, 99, 101, 104, 107, 110, 113, 116, 119, 122, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 171

Offset: 1

Enrique Pérez Herrero, Mar 28 2010

nonn

a(n) is the maximum value on or below diagonal of Kimberling's Expulsion Array; this part could be called the Lower Shuffle.

D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998
R. K. Guy, Unsolved Problems Number Theory, Sect E35.

Enrique Pérez Herrero, Table of n, a(n) for n=1..20000
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991; Solution to Problem 1615, Crux Mathematicorum, Vol. 18, March 1992, p. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence.

Cf. A006852, A007063, A035486, A035505.

(* By direct computation *)
K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
K[i_] := K[i] = K[i, i]; SetAttributes[K, Listable];
A175312[n_] := Max[Table[K[n, i], {i, 1, n}]]  (* Enrique Pérez Herrero, Mar 30 2010 *)
(* By the Formula *)
\[Lambda][n_] := Floor[Log[2, (n + 2)/3]];
A175312[n_] := 1 + 3*(n - \[Lambda][n]) - Floor[(n + 2)/(2^\[Lambda][n])] (* Enrique Pérez Herrero, Mar 30 2010 *)

lambda(n)= floor(log((n + 2)/3)/log(2));
A175312(n)= 1 + 3*(n - lambda(n)) - floor((n + 2)/(2^lambda(n))); \\ Enrique Pérez Herrero, Mar 30 2010

a(n) = 1 + 3(n-lambda(n)) - floor((n+2)/2^lambda(n)), lambda(n) = floor(log_2((n+2)/3)).
a(n) >= A007063(n); a(n) = max(K(n,1),K(n,2),...,K(n,n)), where K(i,j) is an element of Kimberling's Array given by A035486.
From Enrique Pérez Herrero, Mar 30 2010: (Start)
a(theta(k)) = A007063(theta(k)), where theta(k) = Sum_{i=0..k-1} 2^floor(i/3).
At these values the maximum in the Lower Shuffle is the diagonal expelled element. (End)

A356379 Main diagonal of the LORI variant of the array A035486; this is one of eight such sequences discussed in A007063.

Original entry on oeis.org

1, 3, 5, 7, 4, 12, 11, 17, 10, 22, 21, 9, 23, 33, 8, 27, 16, 44, 26, 18, 30, 55, 41, 35, 14, 25, 65, 20, 67, 78, 43, 64, 49, 66, 76, 61, 85, 101, 60, 100, 32, 62, 111, 52, 68, 124, 80, 93, 86, 102, 92, 131, 115, 51, 110, 58, 77, 73, 72, 15, 134, 171, 29, 151

Offset: 1

Clark Kimberling, Oct 21 2022

nonn

Conjecture: every positive integer except 2 occurs exactly once.

Cf. A007063, A035486.

lori = Join[{{1}}, NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
      Range[(Length[#] + 3)/2, Length[#]]]]],
      Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, 200]];
s = Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 150] &[
      ToExpression[#]]} &, {"lori"}];
Last[First[s]]   (* A356379 *)
(* Peter J. C. Moses, Jul 26 2022 *)
(* The next program generates the LORI array. *)
len = 8; lori = Join[{{1}}, NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
     Range[(Length[#] + 3)/2, Length[#]]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
Grid[Map[Flatten, Transpose[{#, Range[3 Range[Length[#]] - 1,
       4 (Length[#] - 2) - 1 + Range[Length[#]]]}]] &[lori]]
(* Peter J. C. Moses, Aug 02 2022 *)

A356377 Main diagonal of the ROLI variant of the array A035486; this is one of eight such sequences discussed in A007063.

Original entry on oeis.org

1, 3, 5, 4, 8, 6, 10, 15, 2, 9, 13, 26, 11, 12, 33, 34, 35, 29, 22, 37, 44, 48, 56, 39, 43, 54, 36, 16, 23, 25, 76, 81, 47, 30, 42, 14, 72, 38, 74, 71, 68, 92, 77, 46, 69, 94, 78, 128, 45, 110, 89, 73, 135, 90, 62, 115, 101, 104, 85, 153, 113, 158, 171, 172

Offset: 1

Clark Kimberling, Oct 21 2022

nonn

Conjecture: every positive integer occurs exactly once.

Cf. A007063, A035486.

len = 8; roli = Join[{{1}}, NestList[Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1], Range[(Length[#] - 1)/2, 1, -1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, 200]];
s = Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 150] &[
      ToExpression[#]]} &, {"roli"}];
Last[First[s]]   (* A356377 *)
(* Peter J. C. Moses, Jul 26 2022 *)
(* The next program generates the ROLI array. *)
len = 8; roli = Join[{{1}}, NestList[Join[#[[Riffle[Range[Length[#], (Length[#] + 3)/2, -1], Range[(Length[#] - 1)/2, 1, -1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
Grid[Map[Flatten, Transpose[{#, Range[3 Range[Length[#]] - 1,
       4 (Length[#] - 2) - 1 + Range[Length[#]]]}]] &[roli]]
(* Peter J. C. Moses, Aug 02 2022 *)

A356378 Main diagonal of the RILO variant of the array A035486; this is one of eight such sequences discussed in A007063.

Original entry on oeis.org

1, 3, 5, 2, 10, 9, 15, 8, 20, 19, 7, 21, 31, 6, 25, 14, 42, 24, 16, 28, 53, 39, 33, 12, 23, 63, 18, 65, 76, 41, 62, 47, 64, 74, 59, 83, 99, 58, 98, 30, 60, 109, 50, 66, 122, 78, 91, 84, 100, 90, 129, 113, 49, 108, 56, 75, 71, 70, 13, 132, 169, 27, 149, 43

Offset: 1

Clark Kimberling, Oct 21 2022

nonn

Conjecture: every positive integer occurs exactly once.

Cf. A007063, A035486.

rilo = Join[{{1}}, NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],          Range[1, (Length[#] - 1)/2, 1]]]], Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, 200]]; (* A356378 *)
s = Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 150] &[
      ToExpression[#]]} &, {"rilo"}];
Last[First[s]]   (* A356378 *)
(* Peter J. C. Moses,Jul 26 2022 *)
(* The next program generates the RILO array. *)
len = 8; rilo = Join[{{1}}, NestList[Join[#[[Riffle[Range[(Length[#] + 3)/2, Length[#]],
     Range[1, (Length[#] - 1)/2, 1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
Grid[Map[Flatten, Transpose[{#, Range[3 Range[Length[#]] - 1,
       4 (Length[#] - 2) - 1 + Range[Length[#]]]}]] &[rilo]]
(* Peter J. C. Moses, Aug 02 2022 *)

A356376 Main diagonal of the LORO variant of the array A035486; this is one of eight such sequences discussed in A007063.

Original entry on oeis.org

1, 3, 5, 6, 4, 11, 12, 9, 13, 15, 23, 7, 27, 16, 24, 25, 34, 36, 19, 14, 50, 41, 10, 40, 60, 32, 43, 35, 26, 20, 38, 63, 79, 81, 57, 44, 74, 80, 65, 72, 107, 28, 53, 93, 76, 66, 114, 56, 129, 55, 119, 47, 103, 125, 85, 39, 45, 141, 106, 77, 98, 137, 109, 33

Offset: 1

Clark Kimberling, Oct 21 2022

nonn

Conjecture: every positive integer except 2 occurs exactly once.

Cf. A007063, A035486.

loro = Join[{{1}}, NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
      Range[Length[#], (Length[#] + 3)/2, -1]]]],
      Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, 200]];
s = Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 150] &[
      ToExpression[#]]} &, {"loro"}]; u = Last[First[s]]
(* Peter J. C. Moses, Jul 26 2022 *)
(* The next program generates the LORO array. *)
len = 8; loro = Join[{{1}}, NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
     Range[Length[#], (Length[#] + 3)/2, -1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
Grid[Map[Flatten, Transpose[{#, Range[3 Range[Length[#]] - 1,
       4 (Length[#] - 2) - 1 + Range[Length[#]]]}]] &[loro]]
(* Peter J. C. Moses, Aug 02 2022 *)

A356380 Main diagonal of the LIRO variant of the array A035486; this is one of eight such sequences discussed in A007063.

Original entry on oeis.org

1, 3, 5, 6, 4, 11, 13, 2, 7, 14, 24, 9, 10, 31, 35, 33, 27, 23, 38, 42, 46, 54, 37, 44, 52, 34, 17, 21, 26, 77, 79, 45, 28, 40, 12, 70, 36, 72, 69, 66, 90, 75, 47, 67, 95, 76, 126, 43, 108, 87, 74, 133, 88, 60, 116, 99, 102, 86, 151, 111, 156, 169, 173, 171

Offset: 1

Clark Kimberling, Oct 24 2022

nonn

Conjecture: every positive integer occurs exactly once.

Cf. A007063, A035486.

lori = Join[{{1}}, NestList[Join[#[[Riffle[Range[1, (Length[#] - 1)/2],
      Range[(Length[#] + 3)/2, Length[#]]]]],
      Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, 200]];
s = Map[{#, Take[Flatten[Map[Take[#, {(Length[#] + 1)/2}] &, #]], 150] &[
      ToExpression[#]]} &, {"lori"}];
Last[First[s]]   (* A356379 *)
(* Peter J. C. Moses, Jul 26 2022 *)
(* The next program generates the LIRO array. *)
len = 8; liro = Join[{{1}}, NestList[Join[#[[Riffle[Range[(Length[#] - 1)/2, 1, -1],
     Range[Length[#], (Length[#] + 3)/2, -1]]]],
     Range[#, # + 2] &[(3 Length[#] + 1)/2]] &, {2, 3, 4}, len]];
Grid[Map[Flatten, Transpose[{#, Range[3 Range[Length[#]] - 1,
       4 (Length[#] - 2) - 1 + Range[Length[#]]]}]] &[liro]]
(* Peter J. C. Moses, Aug 02 2022 *)

A035505 Active part of Kimberling's expulsion array as a triangular array.

Original entry on oeis.org

4, 2, 6, 2, 7, 4, 8, 7, 9, 2, 10, 6, 6, 2, 11, 9, 12, 7, 13, 8, 13, 12, 8, 9, 14, 11, 15, 2, 16, 6, 2, 11, 16, 14, 6, 9, 17, 8, 18, 12, 19, 13, 18, 17, 12, 9, 19, 6, 13, 14, 20, 16, 21, 11, 22, 2, 16, 14, 21, 13, 11, 6, 22, 19, 2, 9, 23, 12, 24, 17, 25, 18, 23, 2, 12, 19, 24, 22, 17, 6

Offset: 1

N. J. A. Sloane

Active or shuffle part of Kimberling's expulsion array (A035486) is given by the elements K(i,j), where j < 2*i-3. [Enrique Pérez Herrero, Apr 14 2010]

			4 2; 6 2 7 4; 8 7 9 2 10 6; ...

R. K. Guy, Unsolved Problems Number Theory, Sect. E35.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991; Solution to Problem 1615, Crux Mathematicorum, Vol. 18, March 1992, p. 82-83.

Cf. A006852, A007063, A038807, A035486.

Cf. A175312, A074294, A000194, A006852, A007063. [Enrique Pérez Herrero, Apr 14 2010]

A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2];
A074294[n_] := n - 2*Binomial[Floor[1/2 + Sqrt[n]], 2];
K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
A035505[n_] := K[A000194[n] + 2, A074294[n]]
(* Enrique Pérez Herrero, Apr 14 2010 *)

From Enrique Pérez Herrero, Apr 14 2010: (Start)
a(n) = K(A000194(n)+2, A074294(n)), where
  K(i,j) = i + j - 1; (j >= 2*i - 3)
  K(i,j) = K(i-1, i-(j+2)/2) if j is even and j < 2*i - 3
  K(i,j) = K(i-1, i+(j-1)/2); if j is odd and j < 2*i - 3.
(End)

More terms from James Sellers, Dec 23 1999

cp's OEIS Frontend

A006852 Step at which n is expelled in Kimberling's puzzle (A035486).

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A007063 Main diagonal of Kimberling's expulsion array (A035486).

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A038807 Future of the smallest-perizeroin komet in Kimberling's expulsion array (A035486).

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A175312 Maximum value on Lower Shuffle Part of Kimberling's Expulsion Array (A035486).

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A356379 Main diagonal of the LORI variant of the array A035486; this is one of eight such sequences discussed in A007063.

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A356377 Main diagonal of the ROLI variant of the array A035486; this is one of eight such sequences discussed in A007063.

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A356378 Main diagonal of the RILO variant of the array A035486; this is one of eight such sequences discussed in A007063.

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A356376 Main diagonal of the LORO variant of the array A035486; this is one of eight such sequences discussed in A007063.

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