1,2

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.

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.

Classic Sequences

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).

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

Cf. A000045, A002878, A035513, A035506.

Sequence in context: A135685 A164658 A079067 this_sequence A065134 A088673 A035614

Adjacent sequences: A160268 A160269 A160270 this_sequence A160272 A160273 A160274

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), May 07 2009