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A242103 Numbers m such that m^2 - 1 is the product of three distinct Fibonacci numbers > 1.

%I A242103 #15 Sep 06 2014 00:40:39
%S A242103 7,9,11,14,29,76,121,199,329,521,659,1364,3571,4523,7307,9349,24476,
%T A242103 64079,167761,212533,439204,1149851,3010349,7881196,20633239,54018521,
%U A242103 141422324,370248451,969323029,2537720636,6643838879,17393796001,45537549124,119218851371
%N A242103 Numbers m such that m^2 - 1 is the product of three distinct Fibonacci numbers > 1.
%C A242103 Conjecture : except the numbers 9, 14, 121, 329, 659, 4523, 7307 and 212533, a(n) is a Lucas number (A000204).
%e A242103 The non-Lucas number 9 is in the sequence because 9^2-1 = 80 = 2*5*8 is the product of three Fibonacci numbers.
%e A242103 The Lucas number 11 is in the sequence because 11^2-1 = 120 = 3*5*8 is the product of three Fibonacci numbers.
%p A242103 with(combinat,fibonacci):with(numtheory):nn:=150:lst:={}:T:=array(1..nn):
%p A242103    for n from 1 to nn do:
%p A242103     T[n]:=fibonacci(n):
%p A242103    od:
%p A242103      for p from 1 to nn-1 do:
%p A242103        for q from p+1 to nn-1 do:
%p A242103           for r from q+1 to nn-1 do:
%p A242103            f:=T[p]*T[q]*T[r]+1:x:=sqrt(f):
%p A242103            if x=floor(x)and T[p]<>1
%p A242103            then
%p A242103            lst:=lst union {x}:
%p A242103            else
%p A242103            fi:
%p A242103           od:
%p A242103        od:
%p A242103      od:
%p A242103      print(lst):
%o A242103 (PARI)
%o A242103 v=[];for(i=3,100,for(j=i+1,100,for(k=j+1,100,s=fibonacci(i)*fibonacci(j)*fibonacci(k);if(issquare(s+1),v=concat(sqrtint(s+1),v)))));v=vecsort(v);v \\ _Derek Orr_, Aug 27 2014
%Y A242103 Cf. A245688, A242074, A000204.
%K A242103 nonn
%O A242103 1,1
%A A242103 _Michel Lagneau_, Aug 15 2014