OFFSET
1,7
COMMENTS
David W. Wilson conjectured (Dec 14 2005) that this sequence consists only of Fibonacci numbers. Proofs were found by Franklin T. Adams-Watters and Don Reble, Dec 14 2005. The following is Reble's proof:
Rearrange A049997, as suggested by Bernardo Boncompagni:
1
2
3 4
5 6
8 9 10
13 15 16
21 24 25 26
34 39 40 42
55 63 64 65 68
89 102 104 105 110
144 165 168 169 170 178
233 267 272 273 275 288
377 432 440 441 442 445 466
Then we know that
F(a+1) * F(a-1) - F(a) * F(a) = (-1)^a
F(a+1) * F(b-1) - F(a-1) * F(b+1)
= + (-1)^b F(a-b), if a>b
= - (-1)^a F(b-a), if a<b
Use these to show that from F(x) to F(x+1), the representable numbers are
F(x) = F(x) * F(2)
< F(x-2) * F(4)
< F(x-4) * F(6)
< ...
< F(x-3) * F(5)
< F(x-1) * F(3)
< F(x+1) * F(1) = F(x+1)
(If x is even, the first identity is needed when the parity changes in the middle.)
Each Fibonacci-product is in one of those subsequences and the identities show that each difference is a Fibonacci number.
LINKS
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35. (Includes a proof of the conjecture proved in Comments.)
MATHEMATICA
t = Take[ Union@Flatten@Table[ Fibonacci[i]Fibonacci[j], {i, 0, 20}, {j, 0, i}], 85]; Drop[t, 1] - Drop[t, -1] (* Robert G. Wilson v, Dec 14 2005 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Dec 14 2005
Name edited by Michel Marcus, Mar 11 2016
STATUS
approved