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A266960
Integer averages of first n Fibonacci numbers (beginning with F(0)).
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?
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
Sequence in context: A057243 A294130 A012470 * A056592 A349681 A282544
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Jan 07 2016
STATUS
approved