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%I #10 Mar 17 2019 21:12:27
%S 1,2,0,3,0,1,2,0,2,1,4,1,3,2,2,3,0,3,3,4,3,5,1,4,4,6,6,5,4,0,4,4,7,9,
%T 10,8,6,1,5,5,8,11,15,16,13,3,0,5,5,9,12,18,24,26,21,5,2,6,6,10,14,20,
%U 29,39,42,34,7,1,5,6,11,15,23,32,47,63,68,55,4,0,6,7,12,17,25,37,52,76,102
%N Monotonic justified array of all positive Fibonacci sequences.
%C Every pair a,b of nonnegative integers occurs in a row. If a>b, then a is in column 1 and b in column 2. The classical Fibonacci sequence (A000045) is in row 1; the Lucas sequence (A002878) is in row 3. Reorderings of the rows and deletions of certain initial terms give the Wythoff array (A035513), the Stolarsky array (A035506), and other arrays in which every positive integer occurs exactly once and every row satisfies the recurrence r(n)=r(n-1)+r(n-2). See the reference for open questions regarding such arrays.
%H Clark Kimberling, <a href="https://doi.org/10.1007/978-94-011-2058-6_39">Orderings of the set of all positive Fibonacci sequences</a>, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. 5 (1993), pp. 405-416.
%H <a href="/classic.html">Classic Sequences</a>
%F Each row begins with integers a,b satisfying a>b>=0.
%F The rows are ordered by the following relation on the first two terms a,b and c,d: (a,b)<(c,d) if and only there exists N such that aF(n)+bF(n+1)<cF(n)+dF(n+1) for every n>=N, where F(n)=A000045(n). In terms of r(1)=a and r(2)=b, the remaining terms of a row are determined by r(n)=r(n-1)+r(n-2).
%e Northwest corner:
%e 1...0...1...1...2...3...5...8..13..21
%e 2...0...2...2...4...6..10..16..26..42
%e 3...0...3...3...6...9..15..24..39..63
%e 2...1...3...4...7..11..18..29..47..76
%Y Cf. A000045, A002878, A035513, A035506.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, May 07 2009
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