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

%I #18 Nov 15 2020 19:17:03

%S 0,1,2,6,13,356,3126,28691,70268,271396,6534495,64591632,162057126,

%T 26237436541,66438353080,7020479040553,11201604625686,296414282891996,

%U 32360305554728271,339791857819043616,871053578019254406,5731478440138170841,9181907843495831675

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

%C It seems only 0, 1, 2, 13 are Fibonacci numbers.

%C Are there other Fibonacci numbers of the form (Fibonacci(k) - 1) / (k - 1)?

%C 2 and 13 are the prime numbers. Are there other prime numbers in this sequence?

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

%e 1 is a term because (Fibonacci(0) + Fibonacci(1) + Fibonacci(2) + Fibonacci(3)) / 4 = 4 / 4 = 1.

%e 2 is a term because (Fibonacci(0) + Fibonacci(1) + Fibonacci(2) + Fibonacci(3) + Fibonacci(4) + Fibonacci(5)) / 6 = 12 / 6 = 2.

%t Table[Mean@ Fibonacci@ Range[0, n], {n, 0, 100}] /. _Rational -> Nothing (* _Michael De Vlieger_, Jan 07 2016 *)

%t 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 *)

%o (PARI) m(n) = sum(k=0, n, fibonacci(k)) % (n+1);

%o b(n) = sum(k=0, n, fibonacci(k)) / (n+1);

%o for(n=0, 1e2, if(m(n)==0, print1(b(n), ", ")));

%Y Cf. A101907, A219612.

%K nonn,easy

%O 1,3

%A _Altug Alkan_, Jan 07 2016