Search: id:A049998
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%I A049998
%S A049998 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,
%T A049998 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,
%U A049998 1,1,8,55,377,233,34,5,1,2,13,89,610,377,55,8,1,1,3,21,144,987,610,89
%N A049998 a(n)=b(n)-b(n-1), where b=A049997 (differences of products of Fibonacci 
               numbers).
%C A049998 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:
%C A049998 Rearrange A049997, as suggested by Bernardo Boncompagni (redgolpe(AT)redgolpe.com):
%C A049998 1
%C A049998 2
%C A049998 3 4
%C A049998 5 6
%C A049998 8 9 10
%C A049998 13 15 16
%C A049998 21 24 25 26
%C A049998 34 39 40 42
%C A049998 55 63 64 65 68
%C A049998 89 102 104 105 110
%C A049998 144 165 168 169 170 178
%C A049998 233 267 272 273 275 288
%C A049998 377 432 440 441 442 445 466
%C A049998 Then we know that
%C A049998 F(a+1) * F(a-1) - F(a) * F(a) = (-1)^a
%C A049998 F(a+1) * F(b-1) - F(a-1) * F(b+1)
%C A049998 = + (-1)^b F(a-b), if a>b
%C A049998 = - (-1)^a F(b-a), if a<b
%C A049998 Use these to show that from F(x) to F(x+1), the representable numbers 
               are
%C A049998 F(x) = F(x) * F(2)
%C A049998 < F(x-2) * F(4)
%C A049998 < F(x-4) * F(6)
%C A049998 < ...
%C A049998 < F(x-3) * F(5)
%C A049998 < F(x-1) * F(3)
%C A049998 < F(x+1) * F(1) = F(x+1)
%C A049998 (If x is even, the first identity is needed when the parity changes in 
               the middle.)
%C A049998 Each Fibonacci-product is in one of those subsequences and the identities 
               show that each difference is a Fibonacci number.
%t A049998 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 *)
%Y A049998 A049997 gives numbers of the form F(i)*F(j), when these Fibonacci-products 
               are arranged in order without duplicates.
%Y A049998 Sequence in context: A137569 A089177 A023996 this_sequence A029253 A016441 
               A131333
%Y A049998 Adjacent sequences: A049995 A049996 A049997 this_sequence A049999 A050000 
               A050001
%K A049998 nonn
%O A049998 1,7
%A A049998 Clark Kimberling (ck6(AT)evansville.edu)
%E A049998 More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Dec 14 2005

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