

A049998


a(n) = b(n)b(n1), where b=A049997 are numbers of the form Fibonacci(i)*Fibonacci(j).


2



1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 3, 1, 1, 8, 5, 1, 2, 13, 8, 1, 1, 3, 21, 13, 2, 1, 5, 34, 21, 3, 1, 1, 8, 55, 34, 5, 1, 2, 13, 89, 55, 8, 1, 1, 3, 21, 144, 89, 13, 2, 1, 5, 34, 233, 144, 21, 3, 1, 1, 8, 55, 377, 233, 34, 5, 1, 2, 13, 89, 610, 377, 55, 8, 1, 1, 3, 21, 144, 987, 610, 89
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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. AdamsWatters 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(a1)  F(a) * F(a) = (1)^a
F(a+1) * F(b1)  F(a1) * F(b+1)
= + (1)^b F(ab), if a>b
=  (1)^a F(ba), 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(x2) * F(4)
< F(x4) * F(6)
< ...
< F(x3) * F(5)
< F(x1) * 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 Fibonacciproduct is in one of those subsequences and the identities show that each difference is a Fibonacci number.


LINKS

Table of n, a(n) for n=1..84.
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 2835. (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

A049997 gives numbers of the form F(i)*F(j), when these Fibonacciproducts are arranged in order without duplicates.
Sequence in context: A266715 A089177 A023996 * A029253 A288165 A016441
Adjacent sequences: A049995 A049996 A049997 * A049999 A050000 A050001


KEYWORD

nonn


AUTHOR

Clark Kimberling


EXTENSIONS

More terms from Robert G. Wilson v, Dec 14 2005
Name edited by Michel Marcus, Mar 11 2016


STATUS

approved



