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A187470 Array: five joint rank sequences tending to A187413, by columns.

%I A187470 #8 Dec 04 2016 19:46:24
%S A187470 1,1,1,1,1,4,3,3,3,3,6,4,5,5,5,9,6,7,7,7,12,8,9,9,9,14,9,11,10,10,17,
%T A187470 11,13,12,13,19,12,15,14,14,22,14,17,16,16,25,16,19,18,18,27,17,21,19,
%U A187470 20,30,19,23,21,22,33,21,25,23,24,35,22,27,25,26,38,24,29,27,28,40,25,31,28,29
%N A187470 Array: five joint rank sequences tending to A187413, by columns.
%C A187470 Precedents are discussed at A187224: adjusted joint rank sequence (AJRS) and the rank transform.
%C A187470 Row 1 (A003622, odds) is the AJRS of the lower Wythoff sequence (A000201) and the natural number sequence, A000027.  Row 2 (A000201) is the AJRS of A000201 and row 1; row 3 (A005408) is the AJRS of A000201 and row 2; etc.  The limit row (not shown) is the rank transform of A000201, which is A187413.  The array shows the first five AJRSs and indicates fairly rapid convergence.
%e A187470 The array consists of five sequences:
%e A187470 1..4..6..9..12..14..17..19..22..25..27..30..33..35..38..40..
%e A187470 1..3..4..6..8...9...11..12..14..16..17..19..21..22..24..25..
%e A187470 1..3..5..7..9...11..13..15..17..19..21..23..25..27..29..31..
%e A187470 1..3..5..7..9...10..12..14..16..18..19..21..23..25..27..28..
%e A187470 1..3..5..7..9...10..13..14..16..18..20..22..24..26..28,,29..
%t A187470 r = (1 + 5^(1/2))/2;
%t A187470 seqA = Table[Floor[r*n], {n, 1, 120}];  (* A000201 *)
%t A187470 seqB = Table[n, {n, 1, 120}]jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
%t A187470 Flatten@Position[#1, {_, 2}]} & [Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@seqB}, 1]]; (#1[[1]] &) /@
%t A187470 FixedPointList[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}], 4];
%t A187470 TableForm[%]
%t A187470 (* by _Peter J. C. Moses_, Mar 10 2011 *)
%Y A187470 A187224, A187469, A187471.
%K A187470 nonn,tabf
%O A187470 1,6
%A A187470 _Clark Kimberling_, Mar 10 2011