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A143514 Array D of denominators of Best Remaining Lower Approximates of x=(1+sqrt(5))/2, by antidiagonals.

%I A143514 #17 Jun 01 2022 12:16:59
%S A143514 1,2,3,5,4,6,13,7,9,8,34,10,12,11,16,89,18,15,14,19,21,233,26,23,17,
%T A143514 22,24,29,610,47,31,20,25,27,32,37,1597,68,39,28,33,30,35,40,42,4181,
%U A143514 123,60,36,41,38,43,48,45,50,10946,178,81,44,49,46,51,56,53,58
%N A143514 Array D of denominators of Best Remaining Lower Approximates of x=(1+sqrt(5))/2, by antidiagonals.
%C A143514 (1) Row 1 of R consists of lower principal convergents to x.
%C A143514 (2) (row limits of R) = x; (column limits of R) = 0.
%C A143514 (3) Every positive integer occurs exactly once in D, so that as a sequence, A143514 is a permutation of the positive integers.
%C A143514 (4) p=floor(q*r) for every p/q in R. Consequently, the terms of N are distinct and their ordered union is the lower Wythoff sequence, A000201.
%C A143514 (5) Conjecture: Every (N(n,k+1)-N(n,k))/(D(n,k+1)-D(n,k)) is a principal convergent to x.
%C A143514 (6) Suppose n>=1 and p/q and s/t are consecutive terms in row n of R. Then (conjecture) q*s-p*t=n.
%C A143514 In general, for irrational r, let {n*r} denote the fractional part of n*r. Define t(1,1) = 1, and t(1,n) = least k such that {k*r} < {t(1,n-1)*x} for n >= 2. Inductively, for m >= 2 and n >= 1, let t(m,1) be the least k not already defined as a term in the array, and for n >= 2, define t(m,n) = least k such that {k*r} < {t(m,n-1)*x and k has not previously been defined as a term. Thus every row of (t(m,n)) is strictly decreasing. For r = (1+sqrt(5))/2, the array (t(m,n)) is D. - _Clark Kimberling_, Feb 21 2021
%H A143514 Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.
%F A143514 For any positive irrational number x, define an array D by successive rows as follows: D(n,k) = least positive integer q not already in D such that there exists an integer p such that 0 < x - p/q < x - c/d for every positive rational number c/d that has 0 < d < q. Thus p/q is the "best remaining lower approximate" of x when all better lower approximates are unavailable. For each q, define N(n,k)=p and R(n,k)=p/q. Then R is the "array of best remaining lower approximates of x," D is the corresponding array of denominators and N, of numerators.
%e A143514 Northwest corner of D:
%e A143514   1 2 5 13
%e A143514   3 4 7 10
%e A143514   6 9 12 15
%e A143514   8 11 14 17
%e A143514 Northwest corner of R:
%e A143514   1/1 3/2 8/5 21/13
%e A143514   4/3 6/4 11/7 16/10
%e A143514   9/6 14/9 19/12 24/15
%e A143514   12/8 17/11 22/14 27/17
%t A143514 r = N[(1 + Sqrt[5])/2, 100]; Table[d = 1; t[k] = {};
%t A143514 Do[a = FractionalPart[n*r];
%t A143514   If[a < d && ! MemberQ[Apply[Union, Map[t[#] &, Range[k - 1]]], n],
%t A143514    d = a; AppendTo[t[k], n]], {n, 10000}]; t[k], {k, 12}];
%t A143514 Column[Table[t[k], {k, 1, 12}]]
%t A143514 (* _Peter J. C. Moses_, Feb 18 2021 *)
%Y A143514 Cf. A000045, A000201, A143515, A143516.
%K A143514 nonn,tabl,frac
%O A143514 1,2
%A A143514 _Clark Kimberling_, Aug 22 2008, Aug 25 2008