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A160271 Monotonic justified array of all positive Fibonacci sequences.

%I A160271 #10 Mar 17 2019 21:12:27
%S A160271 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 A160271 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 A160271 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 A160271 Monotonic justified array of all positive Fibonacci sequences.
%C A160271 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 A160271 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 A160271 <a href="/classic.html">Classic Sequences</a>
%F A160271 Each row begins with integers a,b satisfying a>b>=0.
%F A160271 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 A160271 Northwest corner:
%e A160271 1...0...1...1...2...3...5...8..13..21
%e A160271 2...0...2...2...4...6..10..16..26..42
%e A160271 3...0...3...3...6...9..15..24..39..63
%e A160271 2...1...3...4...7..11..18..29..47..76
%Y A160271 Cf. A000045, A002878, A035513, A035506.
%K A160271 nonn,tabl
%O A160271 1,2
%A A160271 _Clark Kimberling_, May 07 2009