A160271 Monotonic justified array of all positive Fibonacci sequences.

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, 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, 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

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

Table of n, a(n) for n=1..89.

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

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

Cf. A000045, A002878, A035513, A035506.

Sequence in context: A164658 A079067 A356676 * A274912 A065134 A088673

Adjacent sequences: A160268 A160269 A160270 * A160272 A160273 A160274

Keyword

nonn,tabl

Author

Clark Kimberling, May 07 2009

Status

approved