OFFSET
1,2
COMMENTS
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.
LINKS
Clark Kimberling, Orderings of the set of all positive Fibonacci sequences, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. 5 (1993), pp. 405-416.
FORMULA
Each row begins with integers a,b satisfying a>b>=0.
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).
EXAMPLE
Northwest corner:
1...0...1...1...2...3...5...8..13..21
2...0...2...2...4...6..10..16..26..42
3...0...3...3...6...9..15..24..39..63
2...1...3...4...7..11..18..29..47..76
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, May 07 2009
STATUS
approved