A184413 -id:A184413 - 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 *)

A184414 Upper s(n)-Wythoff sequence, where s(n)=floor[(n+1)/2].

Original entry on oeis.org

2, 4, 7, 8, 12, 13, 15, 18, 21, 22, 25, 26, 30, 31, 35, 36, 38, 41, 43, 44, 48, 50, 52, 54, 58, 59, 61, 63, 66, 68, 71, 72, 74, 77, 80, 82, 84, 86, 89, 90, 94, 96, 98, 100, 102, 104, 107, 109, 112, 113, 117, 118, 120, 122, 125, 127, 130, 132, 135, 136, 139, 141, 143, 146, 148, 149, 153, 155, 158, 159, 162, 164, 166, 168, 171, 172, 176, 177, 180, 182, 185, 186, 189, 192, 194, 195, 198, 200, 202, 205, 207, 209, 212, 214, 217, 218, 222, 223, 225, 228

Offset: 1

Clark Kimberling, Jan 13 2011

nonn

See A184117 for the definition of lower and upper s(n)-Wythoff sequences.

Cf. A184413, A184117.

mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=Floor[(n+1)/2];a[1]=1;b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,1,n-1}]]];
Table[s[n],{n,20}]
Table[a[n],{n,100}]
Table[b[n],{n,100}]

A184410 Ranks of (odd i)+j/r when all i+j/r are ranked; r=sqrt(2), i>=0, j>=0. Complement of A184411.

Original entry on oeis.org

3, 5, 8, 11, 12, 15, 16, 20, 21, 25, 26, 27, 31, 32, 33, 38, 39, 40, 44, 46, 47, 48, 52, 54, 55, 57, 61, 63, 64, 66, 69, 71, 73, 74, 76, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 119, 121, 123, 125, 127, 129

Offset: 1

Clark Kimberling, Jan 13 2011

nonn

			Writing (i,j) for i+j/r, the first 7 in the ranking are (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), (0,3); the ranks where i is odd are 3,5 and the ranks where i is even are 1,2,4,6,7.

Cf. A184397, A184411, A184412, A184413, A184414, A184415, A184416.

A184411 Ranks of (even i)+j/r when all i+j/r are ranked; r=sqrt(2), i>=0, j>=0. Complement of A184410.

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 10, 13, 14, 17, 18, 19, 22, 23, 24, 28, 29, 30, 34, 35, 36, 37, 41, 42, 43, 45, 49, 50, 51, 53, 56, 58, 59, 60, 62, 65, 67, 68, 70, 72, 75, 77, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 118, 120

Offset: 1

Clark Kimberling, Jan 13 2011

nonn

			Writing (i,j) for i+j/r, the first 7 in the ranking are (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), (0,3); the ranks where i is odd are 3,5 and the ranks where i is even are 1,2,4,6,7.

Cf. A184397, A184410, A184412, A184413, A184414, A184415, A184416.

cp's OEIS Frontend

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

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A184414 Upper s(n)-Wythoff sequence, where s(n)=floor[(n+1)/2].

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Mathematica

A184410 Ranks of (odd i)+j/r when all i+j/r are ranked; r=sqrt(2), i>=0, j>=0. Complement of A184411.

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A184411 Ranks of (even i)+j/r when all i+j/r are ranked; r=sqrt(2), i>=0, j>=0. Complement of A184410.

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