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.

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A187471 Array: seven joint rank sequences tending to A184413, by columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 6, 5, 5, 5, 5, 5, 5, 8, 6, 6, 6, 6, 6, 6, 11, 8, 9, 9, 9, 9, 9, 13, 9, 10, 10, 10, 10, 10, 15, 11, 12, 11, 11, 11, 11, 18, 12, 14, 14, 14, 14, 14, 20, 14, 16, 15, 16, 16, 16, 23, 16, 18, 17, 17, 17, 17, 25, 17, 20, 19, 19, 19, 19, 27, 19, 21
Offset: 1

Views

Author

Clark Kimberling, Mar 10 2011

Keywords

Comments

Precedents are discussed at A187224: adjusted joint rank sequence (AJRS) and the rank transform.
Let W=A001951, so that W(n)=floor[n*sqrt(2)].
Row 1 of A187471 is the AJRS of W and the natural number sequence, A000027. Row 2 is the AJRS of W and row 1; row 3 is the AJRS of W and row 2; etc. The limit row (not shown) is the rank transform of W, A184413.

Examples

			The array consists of seven sequences:
1..3..6..8..11..13..15..18..20..23..25..27..30..32..35..37..
1..3..5..6..8...9...11..12..14..16..17..19..20..22..24..25..
1..3..5..6..9...10..12..14..16..18..20..21..24..25..28..29..
1..3..5..6..9...10..11..14..15..17..19..20..22..24..26..28..
1..3..5..6..9...10..11..14..16..17..19..21..23..24..27..28..
1..3..5..6..9...10..11..14..16..17..19..20..23..24..26..28..
1..3..5..6..9...10..11..14..16..17..19..20..23..24..27..28..

Crossrefs

Cf. A187224, A187469, A187470.

Programs

  • Mathematica
    r = 2^(1/2);
seqA = Table[Floor[r*n], {n, 1, 120}];  (* A000201 *)
seqB = Table[n, {n, 1, 120}]jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
Flatten@Position[#1, {_, 2}]} & [Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@seqB}, 1]]; (#1[[1]] &) /@
FixedPointList[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}], 6];
TableForm[%]
(* by Peter J. C. Moses, Mar 10 2011 *)

A187476 Rank transform of the sequence floor(3(n-1)/2); complement of A187477.

Original entry on oeis.org

1, 2, 5, 6, 8, 10, 12, 13, 15, 17, 19, 21, 23, 24, 26, 28, 30, 32, 34, 35, 37, 39, 41, 42, 45, 46, 48, 50, 52, 54, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 96, 97, 99, 101, 103, 104, 107, 108, 110, 112, 114, 115, 117, 119
Offset: 1

Views

Author

Clark Kimberling, Mar 10 2011

Keywords

Comments

See A187224.

Crossrefs

Cf A187224, A187477.

Programs

  • Mathematica
    seqA = Table[Floor[3(n-1)/2], {n, 1, 180}]  (* A032766 *)
seqB = Table[n, {n, 1, 80}];           (* A000027 *)
jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
Flatten@Position[#1, {_, 2}]} &[Sort@Flatten[{{#1, 1} & /@ seqA,
{#1, 2} & /@ seqB}, 1]];
limseqU = FixedPoint[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}]][[1]]                                     (* A187476 *)
Complement[Range[Length[seqA]], limseqU]  (* A187477 *)
(* by Peter J. C. Moses, Mar 10 2011 *)

A187570 Rank transform of the sequence ceiling(n/3); complement of A187571.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100, 101, 102, 103, 104, 105, 107, 108, 109, 111, 112, 113, 115, 116
Offset: 1

Views

Author

Clark Kimberling, Mar 11 2011

Keywords

Comments

Appears to be a duplicate of A045749. - R. J. Mathar, Mar 15 2011
The Mathematica programs shown at A187570 and A045749 confirm equality of the first 500 terms. - Clark Kimberling, Apr 02 2011
The sequence of which A187570 is the rank transform is (1,1,1,2,2,2,3,3,3,4,4,4,...), which is (A002264 without the initial three zeros). For a discussion on rank transforms, see A187224.

Crossrefs

Cf. A187224, A187571, A045749, A045750.

Programs

  • Mathematica
    seqA = Table[Ceiling[n/3], {n, 1, 220}]  (*A002264*)
seqB = Table[n, {n, 1, 220}]; (*A000027*)
jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {, 1}], Flatten@Position[#1, {, 2}]} &[Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU=FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (*A187570*)
Complement[Range[Length[seqA]], limseqU]  (*A187571*)
(*by Peter J. C. Moses, Mar 11 2011*)

A187683 Rank transform of the sequence floor(2n/3); complement of A187683.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 11, 13, 14, 15, 17, 18, 20, 22, 23, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 53, 55, 56, 57, 59, 61, 62, 64, 65, 66, 68, 70, 71, 72, 74, 75, 76, 78, 79, 81, 83, 84, 85, 87, 88, 90, 91, 92, 94, 96, 97, 99, 100, 101, 103, 105, 106, 108, 109, 110, 112, 113, 114, 116, 118, 119, 120, 122, 123, 125, 127, 128, 129, 131, 132, 133, 135, 136, 138
Offset: 1

Views

Author

Clark Kimberling, Mar 12 2011

Keywords

Comments

See A187224.

Crossrefs

Cf. A187224, A187684.

Programs

  • Mathematica
    seqA = Table[Floor[2n/3], {n, 1, 220}]
seqB = Table[n, {n, 1, 220}];(*A000027*)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
 FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (*A187683*)
Complement[Range[Length[seqA]], limseqU]  (*A187684*)
(*by Peter J. C. Moses, Mar 12 2011*)

A187685 Rank transform of the sequence floor(4n/3); complement of A187686.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 21, 22, 24, 26, 27, 29, 31, 33, 34, 37, 38, 40, 42, 43, 45, 47, 48, 50, 52, 54, 55, 58, 59, 61, 63, 65, 66, 68, 70, 71, 74, 75, 76, 79, 80, 82, 84, 85, 87, 89, 91, 92, 95, 96, 97, 100, 102, 103, 105, 107, 108, 110, 112, 114, 116, 117, 119, 121, 123, 124, 126, 128, 130, 131, 133, 134, 137, 138, 140, 142, 144
Offset: 1

Views

Author

Clark Kimberling, Mar 12 2011

Keywords

Comments

See A187224.

Crossrefs

Cf. A187224, A187686.

Programs

  • Mathematica
     seqA = Table[Floor[4n/3], {n, 1, 220}]
seqB = Table[n, {n, 1, 220}];(*A000027*)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
 FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (*A187685*)
Complement[Range[Length[seqA]], limseqU]  (*A187686*)
(*by Peter J. C. Moses, Mar 12 2011*)

A187687 Rank transform of the sequence floor(5n/3); complement of A187688.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 12, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 34, 35, 37, 39, 41, 43, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 65, 67, 69, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 90, 92, 94, 96, 97, 99, 101, 103, 105, 107, 108, 111, 113, 114, 116, 118, 120, 122, 124, 125, 127, 130, 131, 133, 135, 137, 139, 141, 143, 145, 146, 148, 150, 152
Offset: 1

Views

Author

Clark Kimberling, Mar 12 2011

Keywords

Comments

See A187224.

Crossrefs

Cf. A187224, A187688.

Programs

  • Mathematica
    seqA = Table[Floor[5n/3], {n, 1, 220}]
seqB = Table[n, {n, 1, 220}];(*A000027*)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
 FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (*A187687*)
Complement[Range[Length[seqA]], limseqU]  (*A187688*)
(*by Peter J. C. Moses, Mar 12 2011*)

A187840 Complement of A187839.

Original entry on oeis.org

2, 5, 6, 10, 11, 13, 16, 18, 20, 23, 25, 27, 29, 32, 34, 36, 39, 40, 43, 45, 48, 49, 53, 54, 57, 59, 61, 64, 66, 67, 70, 72, 75, 77, 79, 81, 84, 87, 88, 91, 93, 95, 97, 100, 102, 105, 106, 109, 110, 113, 115, 118, 120, 122, 125, 127, 129, 131, 133, 136, 138, 141, 143, 145, 148, 149, 152, 154, 157, 158, 161, 163, 165, 168, 170
Offset: 1

Views

Author

Clark Kimberling, Mar 13 2011

Keywords

Comments

See A187224.

Links

Crossrefs

Cf. A187224, A187839.

Programs

  • Mathematica
    r=2^(1/2);
seqA = Table[Floor[r*n-1/2], {n, 1, 220}]
seqB = Table[n, {n, 1, 220}]; (* A000027 *)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (* A187839 *)
Complement[Range[Length[seqA]], limseqU]  (* A187840 *)
(* Peter J. C. Moses, Mar 13 2011 *)

A187895 Rank transform of the sequence A115384; complement of A187896.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 33, 35, 37, 38, 40, 41, 42, 43, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 81, 82, 84, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 106, 107, 108, 109, 111, 113, 114, 116, 117, 118, 120, 121, 122, 123, 125
Offset: 1

Views

Author

Clark Kimberling, Mar 15 2011

Keywords

Comments

A187895 is the rank transform (see A187224) of the sequence of partial sums of the Thue-Morse sequence A010060.

Crossrefs

Cf. A187224, A187896, A187897.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 8]; t
seqA = Table[Sum[t[[k]], {k, 1, n}], {n, 1, 256}]   (* A115384 *)
seqB = Table[n, {n, 1, 220}]; (* A000027 *)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]]   (* A187895 *)
Complement[Range[Length[seqA]], limseqU]  (* A187896 *)
(* by Peter J. C. Moses,  Mar 15 2011 *)

A187897 Rank transform of the sequence A159481; complement of A187898.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 56, 57, 58, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 74, 75, 76, 77, 79, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 111, 112, 113, 114, 115, 117, 119, 120, 121, 123, 124, 125
Offset: 1

Views

Author

Clark Kimberling, Mar 15 2011

Keywords

Comments

A187897 is the rank transform (see A187224) of the sequence of partial sums of the Thue-Morse sequence A010059.

Crossrefs

Cf. A187224, A187898, A187895.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {1 -> {1, 0}, 0 -> {0, 1}}] &, {1}, 8]; t
seqA = Table[Sum[t[[k]], {k, 1, n}], {n, 1, 256}]   (* A159481 *)
seqB = Table[n, {n, 1, 220}]; (* A000027 *)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
FixedPoint[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}]][[
  1]]   (* A187897 *)
Complement[Range[Length[seqA]], limseqU]  (* A187898 *)
(* by Peter J. C. Moses,  Mar 15 2011 *)

A187907 Rank transform of the sequence floor((4 - sqrt(5))*n); complement of A187908.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 16, 19, 20, 23, 24, 26, 28, 30, 32, 34, 36, 38, 40, 41, 43, 46, 48, 49, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 71, 73, 74, 76, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 139, 141, 143, 145, 147, 149, 151, 153, 155
Offset: 1

Views

Author

Clark Kimberling, Mar 15 2011

Keywords

Comments

See A187224. A187232(n) = A187907(n) for n=1..20; A187232(21)=39 and A187907(21)=40.

Crossrefs

Cf. A187224, A187908.

Programs

  • Mathematica
    r=4-5^(1/2);
seqA = Table[Floor[r*n], {n, 1, 220}] (* A187330 *)
seqB = Table[n, {n, 1, 220}];  (* A000027 *)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (* A187907 *)
Complement[Range[Length[seqA]], limseqU]  (* A187908 *)
(* Peter J. C. Moses, Mar 15 2011 *)
  • Maxima
    makelist(floor((4-sqrt(5))*n),n,1,100); /* Martin Ettl, Oct 17 2012 */
Previous Showing 11-20 of 107 results. Next

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