A259503 - OEIS

%I #17 Aug 04 2015 03:04:11

%S 0,1,2,3,5,12,133

%N Numbers k such that k^2+1 is the product of a Fibonacci number and a Lucas number.

%C Conjecture: the sequence is finite.

%C No more terms below 25*10^4. - _Robert G. Wilson v_, Jul 06 2015

%C No more terms below 10^7. - _Manfred Scheucher_, Aug 03 2015

%H Manfred Scheucher, <a href="/A259503/a259503.sage.txt">Sage Script</a>

%e 0^2+1 = 1 = 1*1 = F(1)*L(1);

%e 1^2+1 = 2 = 2*1 = F(3)*L(1);

%e 2^2+1 = 5 = 5*1 = F(5)*L(1);

%e 3^2+1 = 10 = 5*2 = F(5)*L(0);

%e 5^2+1 = 26 = 13*2 = F(7)*L(0);

%e 12^2+1 = 145 = 5*29 = F(5)*L(7);

%e 133^2+1 = 17690 = 610*29 = F(15)*L(7).

%p with(combinat,fibonacci):nn:=200:lst:={}:

%p a:=n->2*fibonacci(n-1)+fibonacci(n):

%p for i from 0 to nn do:

%p for j from 0 to nn do:

%p   x:=sqrt(a(i)*fibonacci(j)-1):

%p   if x=floor(x) then lst:=lst union {x}:

%p   else fi:

%p od:

%p od:

%p print(lst):

%t fibQ[n_] := (Fibonacci@ Round@ Log[ GoldenRatio, n*Sqrt@ 5 == n); fQ[n_] := Block[{k = 0, l}, While[l = LucasL@ k; l < n^2 + 2 && ! fibQ[(n^2 + 1)/l], k++]; If[l < 2 + n^2, True, False]]; k = 0; lst = {}; While[k < 250001, If[ fQ@ k, AppendTo[lst, k]; Print[k]]; k++]; lst (* _Robert G. Wilson v_, Jul 06 2015 *)

%Y Cf. A000032, A000045, A002522.

%K nonn,more

%O 1,3

%A _Michel Lagneau_, Jun 29 2015