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A111075 a(n) = F(n) * Sum_{k|n} 1/F(k), where F(k) is the k-th Fibonacci number.

%I A111075 #27 Aug 20 2019 04:52:22
%S A111075 1,2,3,7,6,21,14,50,52,122,90,427,234,784,1038,2351,1598,6860,4182,
%T A111075 17262,17262,35622,28658,139703,90031,243308,300405,766850,514230,
%U A111075 2367006,1346270,5188658,5326470,11409346,11782764,44717548,24157818
%N A111075 a(n) = F(n) * Sum_{k|n} 1/F(k), where F(k) is the k-th Fibonacci number.
%C A111075 a(n) = a(n+1) for n = 20, but for no other n < 25000. - _Klaus Brockhaus_, Oct 11 2005
%C A111075 If k|n then F(k)|F(n). Therefore A111075(n) = F(n) * sum{k|n} 1/F(k) = sum{k|n} F(n)/F(k) is a sum of integers. - _Max Alekseyev_, Oct 22 2005
%H A111075 Alois P. Heinz, <a href="/A111075/b111075.txt">Table of n, a(n) for n = 1..4785</a>
%F A111075 G.f.: Sum_{n>=1} x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n). [_Paul D. Hanna_, Jan 09 2012]
%e A111075 a(6) = F(6) sum{k|6} 1/F(k) = F(6) * (1/F(1) + 1/F(2) + 1/F(3) + 1/F(6)) = 8 * (1/1 + 1/1 + 1/2 + 1/8) = 21.
%p A111075 with(combinat): with(numtheory): a:=proc(n) local div: div:=divisors(n): fibonacci(n)*sum(1/fibonacci(div[j]),j=1..tau(n)) end: seq(a(n),n=1..40); # _Emeric Deutsch_, Oct 11 2005
%p A111075 # second Maple program:
%p A111075 a:= n-> (F-> F(n)*add(1/F(d),d=numtheory[divisors(n)))(
%p A111075          combinat[fibonacci]):
%p A111075 seq(a(n), n=1..42);  # _Alois P. Heinz_, Aug 20 2019
%t A111075 f[n_] := Fibonacci[n]*Plus @@ (1/Fibonacci /@ Divisors[n]); Table[ f[n], {n, 37}] (* _Robert G. Wilson v_, Oct 11 2005 *)
%o A111075 (PARI) {for(n=1,37,d=divisors(n);print1(fibonacci(n)*sum(j=1,length(d), 1/fibonacci(d[j])),","))}
%o A111075 (PARI) {a(n)=fibonacci(n) * sumdiv(n,d, 1/fibonacci(d))} /* _Paul D. Hanna_, Oct 11 2005 */
%o A111075 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A111075 {a(n)=polcoeff(sum(m=1,n, x^m/(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))),n)} /* _Paul D. Hanna_, Oct 11 2005 */
%Y A111075 Cf. A000045, A007435, A111159, A000204 (Lucas), A203318.
%K A111075 nonn
%O A111075 1,2
%A A111075 _Leroy Quet_, Oct 10 2005
%E A111075 More terms from _Robert G. Wilson v_, _Emeric Deutsch_, _Paul D. Hanna_ and _Klaus Brockhaus_, Oct 11 2005