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

Author

Clark Kimberling, May 07 2009

Keywords

• nonn
• tabl

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.

Examples

			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
		

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.
• Classic Sequences

Crossrefs

Cf. A000045, A002878, A035513, A035506.

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)=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).