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Averages of the Fibonacci numbers which take integer values.
1

%I #4 Mar 31 2012 10:31:25

%S 1,1,5058,262213938,18124577012898,187952389930860,

%T 1409394295257361938,116903055445824294157698,

%U 10100618828005365858877129458,81435914480042681825934186407384633298,7505278652741640947693896415563573183203138,700346071081054203480884565881868806176873272498

%N Averages of the Fibonacci numbers which take integer values.

%C The numbers n such that F(1)+F(2)+...+F(n)=F(n+2)-1 is divisible by n are given in A111035. [From _Max Alekseyev_, Jun 04 2009]

%F 1/n*Sum {j=1..n} Fibonacci_j is an integer.

%F a(n) = (A000045(A111035(n)+2)-1) / A111035(n) = A000071(A111035(n)+2) / A111035(n) [From _Max Alekseyev_, Jun 04 2009]

%t lst = {}; Do[a = Sum[ Fibonacci@ j, {j, n}]/n; If[ IntegerQ@ a, AppendTo[lst, a]], {n, 250}]; lst

%Y Cf. A050248, integer average of n primes for some n, A000045.

%K easy,nonn

%O 1,3

%A _Daniel Tisdale_, May 25 2009

%E Corrected and extended by _Max Alekseyev_ and _Robert G. Wilson v_, Jun 04 2009