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

7, 9, 11, 14, 29, 76, 121, 199, 329, 521, 659, 1364, 3571, 4523, 7307, 9349, 24476, 64079, 167761, 212533, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371

Offset

1,1

Comments

Conjecture : except the numbers 9, 14, 121, 329, 659, 4523, 7307 and 212533, a(n) is a Lucas number (A000204).

Links

Table of n, a(n) for n=1..34.

Example

The non-Lucas number 9 is in the sequence because 9^2-1 = 80 = 2*5*8 is the product of three Fibonacci numbers.
The Lucas number 11 is in the sequence because 11^2-1 = 120 = 3*5*8 is the product of three Fibonacci numbers.

Maple

with(combinat, fibonacci):with(numtheory):nn:=150:lst:={}:T:=array(1..nn):
   for n from 1 to nn do:
    T[n]:=fibonacci(n):
   od:
     for p from 1 to nn-1 do:
       for q from p+1 to nn-1 do:
          for r from q+1 to nn-1 do:
           f:=T[p]*T[q]*T[r]+1:x:=sqrt(f):
           if x=floor(x)and T[p]<>1
           then
           lst:=lst union {x}:
           else
           fi:
          od:
       od:
     od:
     print(lst):

Prog

(PARI)
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

Crossrefs

Cf. A245688, A242074, A000204.

Sequence in context: A267970 A024828 A139058 * A328454 A254937 A222947

Adjacent sequences: A242100 A242101 A242102 * A242104 A242105 A242106

Keyword

nonn

Author

Michel Lagneau, Aug 15 2014

Status

approved