A187470 -id:A187470 - cp's OEIS frontend

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

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

Clark Kimberling, Mar 10 2011

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.

			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..

Cf. A187224, A187469, A187470.

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 *)

A187469 Array: five joint rank sequences tending to lower Wythoff sequence A000201, by columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 5, 4, 4, 4, 4, 7, 6, 6, 6, 6, 9, 7, 8, 8, 8, 11, 9, 9, 9, 9, 13, 10, 11, 11, 11, 15, 12, 13, 12, 12, 17, 13, 14, 14, 14, 19, 15, 16, 16, 16, 21, 16, 18, 17, 17, 23, 18, 19, 19, 19, 25, 19, 21, 20, 21, 27, 21, 23, 22, 22, 29, 22, 24, 24, 24, 31, 24, 26, 25, 25

Offset: 1

Clark Kimberling, Mar 10 2011

Precedents are discussed at A187224: adjusted joint rank sequence (AJRS) and the rank transform.

Row 1 (A005498, odds) is the AJRS of the natural number sequence N=A000027 with itself. Row 2 is the AJRS of N and row 1; row 3 is the AJRS of N and row 2; etc. The limit row is the rank transform of N, the lower Wythoff sequence, A000201. The array shows the first five AJRSs and indicates fairly rapid convergence.

			The array consists of five sequences:
1..3..5..7..9..11..13..15..17..19..21..23..25..27..29..31..
1..3..4..6..7..9...10..12..13..15..16..18..19..21..22..24..
1..3..4..6..8..9...11..13..14..16..18..19..21..23..24..26..
1..3..4..5..8..9...11..12..14..16..17..19..20..22..24..25..
1..3..4..6..8..9...11..12..14..16..17..19..21..22..24..25..

A187224, A187470, A187471.

seqA = Table[n, {n, 1, 120}];
seqB = seqA;
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}], 4];
TableForm[%]
(* by Peter J. C. Moses, Mar 10 2011 *)

cp's OEIS Frontend

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

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