A049998 a(n) = b(n)-b(n-1), where b=A049997 are numbers of the form Fibonacci(i)*Fibonacci(j).

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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

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

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.)

Crossrefs

A049997 gives numbers of the form F(i)*F(j), when these Fibonacci-products are arranged in order without duplicates.

A049999(n) gives the smallest index k such that Fibonacci(k) equals a(n).

Programs

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 *)

Extensions

More terms from Robert G. Wilson v, Dec 14 2005

Name edited by Michel Marcus, Mar 11 2016

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