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 A111075 a(n) = F(n) * Sum_{k|n} 1/F(k), where F(k) is the k-th Fibonacci number. 5
 1, 2, 3, 7, 6, 21, 14, 50, 52, 122, 90, 427, 234, 784, 1038, 2351, 1598, 6860, 4182, 17262, 17262, 35622, 28658, 139703, 90031, 243308, 300405, 766850, 514230, 2367006, 1346270, 5188658, 5326470, 11409346, 11782764, 44717548, 24157818 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = a(n+1) for n = 20, but for no other n < 25000. - Klaus Brockhaus, Oct 11 2005 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..4785 FORMULA 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] EXAMPLE 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. MAPLE 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 # second Maple program: a:= n-> (F-> F(n)*add(1/F(d), d=numtheory[divisors(n)))(          combinat[fibonacci]): seq(a(n), n=1..42);  # Alois P. Heinz, Aug 20 2019 MATHEMATICA f[n_] := Fibonacci[n]*Plus @@ (1/Fibonacci /@ Divisors[n]); Table[ f[n], {n, 37}] (* Robert G. Wilson v, Oct 11 2005 *) PROG (PARI) {for(n=1, 37, d=divisors(n); print1(fibonacci(n)*sum(j=1, length(d), 1/fibonacci(d[j])), ", "))} (PARI) {a(n)=fibonacci(n) * sumdiv(n, d, 1/fibonacci(d))} /* Paul D. Hanna, Oct 11 2005 */ (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)} {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 */ CROSSREFS Cf. A000045, A007435, A111159, A000204 (Lucas), A203318. Sequence in context: A187015 A245467 A070964 * A011372 A104955 A011161 Adjacent sequences:  A111072 A111073 A111074 * A111076 A111077 A111078 KEYWORD nonn AUTHOR Leroy Quet, Oct 10 2005 EXTENSIONS More terms from Robert G. Wilson v, Emeric Deutsch, Paul D. Hanna and Klaus Brockhaus, Oct 11 2005 STATUS approved

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Last modified December 5 11:56 EST 2021. Contains 349557 sequences. (Running on oeis4.)