A111075 a(n) = F(n) * Sum_{k|n} 1/F(k), where F(k) is the k-th Fibonacci number.
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
Offset: 1
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..4785
Programs
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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
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Mathematica
f[n_] := Fibonacci[n]*Plus @@ (1/Fibonacci /@ Divisors[n]); Table[ f[n], {n, 37}] (* Robert G. Wilson v, Oct 11 2005 *) -
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 */
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]
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