A175312 -id:A175312 - 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

A035486 Kimberling's expulsion array read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 2, 6, 5, 5, 5, 2, 8, 6, 6, 6, 7, 7, 6, 7, 7, 7, 4, 9, 2, 13, 8, 8, 8, 8, 2, 11, 12, 2, 9, 9, 9, 9, 10, 9, 8, 11, 18, 10, 10, 10, 10, 6, 12, 9, 16, 17, 16, 11, 11, 11, 11, 11, 7, 14, 14, 12, 14, 23, 12, 12, 12, 12, 12, 13, 11, 6, 9, 21, 2, 13, 13, 13, 13, 13, 13, 8, 15

Offset: 1

N. J. A. Sloane

To get the next row, start with the first element to the right of the diagonal term, then take the first to the left of the diagonal, then the second to the right, then the second to the left, the third to the right, etc.

It is conjectured since 1992 that the main diagonal elements (A007063) are a permutation of the positive integers.

			The array starts (with elements of A007063 in brackets):
  [1]  2   3   4   5   6   7   8   9  10  11  12 ...
   2  [3]  4   5   6   7   8   9  10  11  12  13 ...
   4   2  [5]  6   7   8   9  10  11  12  13  14 ...
   6   2   7  [4]  8   9  10  11  12  13  14  15 ...
   8   7   9   2 [10]  6  11  12  13  14  15  16 ...
   6   2  11   9  12  [7] 13   8  14  15  16  17 ...
  13  12   8   9  14  11 [15]  2  16   6  17  18 ...
2 occurs as diagonal element in row 25, 27 in row 7598, and 19 in row 49595 (cf. A006852).

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

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.
Enrique Pérez Herrero, Formulas and programs for Kimberling's expulsion array
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, p. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence

Cf. A006852 (positions), A007063 (main diagonal), A035505 (active part), A038807.

Cf. A175312 (maximum value on lower shuffle part).

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];
T[n_] := n*(n + 1)/2;
S[n_] := Floor[1/2 (1 + Sqrt[1 + 8 (n - 1)])];
AJ[n_] := 1 + T[S[n]] - n;
AI[n_] := 1 + S[n] - AJ[n];
A035486[n_] := K[AI[n], AJ[n]];
(* Enrique Pérez Herrero, Mar 30 2010 *)

def A035486(n,k):
    if k >= 2*n-3: return n+k-1
    q,r = divmod(k+1,2)
    return A035486(n-1,n-1+(1-2*r)*q) # Pontus von Brömssen, Jan 28 2023

More terms from James Sellers, Dec 23 1999

Edited by Georg Fischer, Jul 03 2020

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

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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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A035486 Kimberling's expulsion array read by antidiagonals.

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

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