cp's OEIS Frontend

A187224 Rank transform of the sequence floor(3*n/2).

%I A187224 #9 Dec 04 2016 19:46:24
%S A187224 1,3,5,7,8,11,12,14,16,18,19,21,23,25,27,29,30,32,34,36,38,40,41,43,
%T A187224 45,47,48,51,52,54,56,58,60,61,63,65,67,69,70,72,74,76,78,80,81,83,85,
%U A187224 87,89,91,92,94,96,98,100,102,103,105,107,109,110,113,114,116,118,120,121,123,125,127,129,131,132,135,136,138,140,142,143,145,147,149,151,153,154,156,158,160,162,163,165,167,169,171,172,175,176,178,180,182,183,185,187,189,191,193,194,196,198,200
%N A187224 Rank transform of the sequence floor(3*n/2).
%C A187224 Complement of A187225.
%C A187224 The notion of the rank transform of a sequence is introduced as follows.  Suppose that a=(a(n)), for n>=1, is a nondecreasing sequence of nonnegative integers, where a(1)<=1, and suppose that b=(b(n)), for n>=1, is an increasing sequence of positive integers.
%C A187224 Define h(1)=a(1), and for n>1, define h(n)=the number of numbers b(i) satisfying a(n-1)<=b(i)<a(n).
%C A187224 Define r(1)=1, and for n>1, define r(n)=b(n-1)+h(n)+1.
%C A187224 The sequence r is the adjusted rank sequence when a and b are jointly ranked, with a(i) before b(j) when a(i)=b(j).  (For a discussion of adjusted joint rank sequences, see A186219 and A186350.)
%C A187224 If r(n)=b(n) for all n>=1, we call r the rank transform of a and denote it by R(a).  To summarize,
%C A187224   (1)  initial values: r(1)=1, h(1)=a(1);
%C A187224   (2)  counting function: h(n)= # r(i) in the half-open
%C A187224        interval [a(n-1),a(n));
%C A187224   (3)  recurrence:  r(n)=r(n-1)+h(n)+1.
%C A187224 Assuming a unbounded, let c be the number of a(i)<=1, let c(1)=c+1, and for n>1, let c(n) be the rank of r(n) when all the numbers a(i)<=r(n) and r(1),...,r(n-1), r(n) are jointly ranked.  Then, clearly, a(n)<=r(n)<=c(n) for n>=1, and the sequences r and c are a complementary pair.
%C A187224 What conditions on the sequence a will ensure that R(a) exists?  That is, what conditions will ensure that the counting function in (2) can be determined inductively, so that the recurrence (3) can be used to self-generate the sequence r?  The answer is this:  a(n)<=c(n-1)+1; viz., if a(n)>c(n-1)+1, then c(n-1)+1=r(n), but then a(n)>r(n), a contradiction, but if a(n)<=c(n-1)+1, there is no such obstacle.
%C A187224 Examples:
%C A187224 R(A000012)=A000027
%C A187224 R(A000027)=A000201, the lower Wythoff sequence
%C A187224 R(A004526)=A026367
%C A187224 R(A005408)=A005408
%C A187224 Returning now to a and b as above, let (r(1,k)) be the adjusted joint rank sequence (AJRS) of a and b, with a(i) before b(j) when a(i)=b(j).  Let (r(2,k)) be the AJRS of a and (r(1,k)); and inductively, let (r(n,k)) be the AJRS of a and (r(n-1,k)).  If R(a) exists, then the limit of (r(n,k)) is R(a).
%C A187224 Thus, any choice of initial sequence b can be used to determine the first thousand terms of R(a).  In the Mathematica program below, b=(1,2,3,4,...)=A000027.
%e A187224 a... 1..3..4..6..7...9...10..12..13..15..16..18..19..
%e A187224 r... 1..3..5..7..8...11..12..14..16..18..19..21..23..
%e A187224 c... 2..4..6..9..10..13..15..17..20..22..24..26..28..
%e A187224 h... 1..1..1..1..0...2...0...1...1...1...0...1...1...
%e A187224 The sequences which converge to R(a), starting with
%e A187224 a=A187224 and b=A000027:
%e A187224 a(k)....1..3..4..6..7...9...10..12..13..15...
%e A187224 b(k)....1..2..3..4..5...6...7...8...9...10...
%e A187224 r(1,k)..1..4..6..9..11..14..16..19..21..24...
%e A187224 r(2,k)..1..3..4..6..8...9...11..13..14..16...
%e A187224 r(3,k)..1..3..5..7..9...11..13..15..16..19...
%e A187224 r(4,k)..1..3..5..7..8...10..12..14..15..17...
%e A187224 r(5,k)..1..3..5..7..8...11..12..14..16..18...
%t A187224 seqA=Table[Floor[3*n/2], {n,1,220}]     (* A032766 *)
%t A187224 seqB=Table[n, {n,1,120}];               (* A000027 *)
%t A187224 jointRank[{seqA_,seqB_}]:={Flatten@Position[#1,{_,1}],Flatten@Position[#1,{_,2}]}&[Sort@Flatten[{{#1,1}&/@seqA,{#1,2}&/@seqB},1]];
%t A187224 limseqU=FixedPoint[jointRank[{seqA,#1[[1]]}]&,jointRank[{seqA,seqB}]][[1]]                     (* A187224 *)
%t A187224 Complement[Range[Length[seqA]],limseqU] (* A187225 *)
%t A187224 (* by _Peter J. C. Moses_, Mar 07 2011 *)
%Y A187224 Cf. A186219, A186350, A187225.
%K A187224 nonn
%O A187224 1,2
%A A187224 _Clark Kimberling_, Mar 07 2011