A266960 Integer averages of first n Fibonacci numbers (beginning with F(0)).

0, 1, 2, 6, 13, 356, 3126, 28691, 70268, 271396, 6534495, 64591632, 162057126, 26237436541, 66438353080, 7020479040553, 11201604625686, 296414282891996, 32360305554728271, 339791857819043616, 871053578019254406, 5731478440138170841, 9181907843495831675

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?

Links

Table of n, a(n) for n=1..23.

Formula

a(n) = A000071(A219612(n) + 1) / A219612(n).

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

Cf. A101907, A219612.

Sequence in context: A057243 A294130 A012470 * A056592 A349681 A282544

Adjacent sequences: A266957 A266958 A266959 * A266961 A266962 A266963

Keyword

nonn,easy

Author

Altug Alkan, Jan 07 2016

Status

approved