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A245688 Numbers n such that n^2 + 1 is the product of three distinct Fibonacci numbers > 1.

%I A245688 #22 Aug 15 2014 10:25:35
%S A245688 47,99,123,322,843,2207,5778,15075,15127,39603,103682,271443,710647,
%T A245688 1860498,4870847,12752043,33385282,87403803,228826127,599074578,
%U A245688 1568397607,4106118243,10749957122,28143753123,73681302247,192900153618,505019158607,1322157322203
%N A245688 Numbers n such that n^2 + 1 is the product of three distinct Fibonacci numbers > 1.
%C A245688 Conjecture: except the numbers 99 and 15075, all the terms belong to A005248 (bisection of Lucas numbers).
%C A245688 The above conjecture holds for the first 1000 terms. - _Jens Kruse Andersen_, Aug 10 2014
%H A245688 Jens Kruse Andersen, <a href="/A245688/b245688.txt">Table of n, a(n) for n = 1..1000</a>
%F A245688 Empirical g.f.: x*(9297*x^9-24320*x^8-52*x^7+52*x^3-127*x^2-42*x+47) / (x^2-3*x+1). - _Colin Barker_, Aug 13 2014
%e A245688 99 is in the sequence because 99^2+1 = 9802 = 2*13*377 where 2, 13 and 377 are three Fibonacci numbers, but 99 is not a Lucas number.
%e A245688 15075 is in the sequence because 15075^2+1 = 13*89*196418 where 13, 89 and 196418 are three Fibonacci numbers, but 15075 is not a Lucas number.
%p A245688 with(combinat,fibonacci):with(numtheory):nn:=200:lst:={}:T:=array(1..nn):
%p A245688    for n from 1 to nn do:
%p A245688     T[n]:=fibonacci(n):
%p A245688    od:
%p A245688      for p from 1 to nn-1 do:
%p A245688        for q from p+1 to nn-1 do:
%p A245688           for r from q+1 to nn-1 do:
%p A245688            f:=T[p]*T[q]*T[r]-1:x:=sqrt(f):
%p A245688            if x=floor(x)and T[p]<>1
%p A245688            then
%p A245688            lst:=lst union {x}:
%p A245688            else
%p A245688            fi:
%p A245688           od:
%p A245688        od:
%p A245688      od:
%p A245688      print(lst):
%Y A245688 Cf. A000032, A000045, A005248, A245236.
%K A245688 nonn,easy
%O A245688 1,1
%A A245688 _Michel Lagneau_, Jul 29 2014
%E A245688 Doubtful link, formula and PARI code deleted by _Colin Barker_, Jul 31 2014