OFFSET
1,3
COMMENTS
It seems only 0, 1, 2, 13 are Fibonacci numbers.
Are there other Fibonacci numbers of the form (Fibonacci(k) - 1) / (k - 1)?
2 and 13 are the prime numbers. Are there other prime numbers in this sequence?
EXAMPLE
1 is a term because (Fibonacci(0) + Fibonacci(1) + Fibonacci(2) + Fibonacci(3)) / 4 = 4 / 4 = 1.
2 is a term because (Fibonacci(0) + Fibonacci(1) + Fibonacci(2) + Fibonacci(3) + Fibonacci(4) + Fibonacci(5)) / 6 = 12 / 6 = 2.
MATHEMATICA
Table[Mean@ Fibonacci@ Range[0, n], {n, 0, 100}] /. _Rational -> Nothing (* Michael De Vlieger, Jan 07 2016 *)
Module[{nn=100, fibs}, fibs=Accumulate[Fibonacci[Range[0, nn]]]; Select[ #[[1]] / #[[2]]&/@Thread[{fibs, Range[nn+1]}], IntegerQ]] (* Harvey P. Dale, Nov 15 2020 *)
PROG
(PARI) m(n) = sum(k=0, n, fibonacci(k)) % (n+1);
b(n) = sum(k=0, n, fibonacci(k)) / (n+1);
for(n=0, 1e2, if(m(n)==0, print1(b(n), ", ")));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Jan 07 2016
STATUS
approved