cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A355323 Numbers k such that A007063(k) = A356026(k).

Original entry on oeis.org

1, 2, 3, 371, 5131, 9250, 26664
Offset: 1

Views

Author

Clark Kimberling, Jul 25 2022

Keywords

Comments

It is conjectured that this is an infinite sequence.
No more terms < 1000000. - Vaclav Kotesovec, Jan 26 2023

Crossrefs

Cf. A007063, A356026.

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

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

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)

Examples

			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, ... )

References

  • 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).

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

More terms from James Sellers, Dec 23 1999

A282348 Numbers expelled during the formation of the array A282347.

Original entry on oeis.org

1, 3, 5, 2, 8, 9, 4, 10, 7, 20, 12, 24, 14, 23, 29, 31, 18, 28, 22, 47, 39, 38, 46, 57, 11, 41, 27, 6, 55, 66, 63, 76, 69, 59, 15, 34, 53, 70, 79, 104, 99, 48, 97, 58, 89, 25, 13, 126, 87, 30, 115, 82, 91, 52, 143, 67, 95, 75, 123, 64, 37, 86, 44, 169, 88, 113
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2017

Keywords

Crossrefs

Cf. A007063, A282347, A356376, A356026, A356377, A356378, A356379, A356380.

Programs

  • Mathematica
    f[seq_] := Riffle[Take[Reverse[seq], #], Take[seq, #]] &[Floor[Length[seq]/2]];
rows = 200; row[1] = Table[n, {n, rows}];
Table[row[n + 1] = Flatten[{f[Take[row[n], 2 n - 1]], Drop[row[n], 2 n - 1]}], {n, Floor[(rows - 1)/3 + 1]}];
TableForm[Table[Take[row[n], 20], {n, 1, 20}]] (* A282347 *)
Table[row[n][[n]], {n, 2 + Floor[(rows - 1)/3]}] (* A282348 *)
Showing 1-3 of 3 results.

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