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A111075
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F(n) * sum{k|n} 1/F(k), where F(k) is the k-th Fibonacci number.
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5
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) = a(n+1) for n = 20, but for no other n < 25000. - Klaus Brockhaus
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 (maxale(AT)gmail.com), Oct 22 2005
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FORMULA
| G.f.: Sum_{n>=1} x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n). [From Paul D. Hanna, Jan 09 2012]
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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.
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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); (Deutsch)
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MATHEMATICA
| f[n_] := Fibonacci[n]*Plus @@ (1/Fibonacci /@ Divisors[n]); Table[ f[n], {n, 37}] (* Robert G. Wilson v *)
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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 */
(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 */
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CROSSREFS
| Cf. A000045, A111159, A000204 (Lucas), A203318.
Sequence in context: A019585 A187015 A070964 * A011372 A104955 A011161
Adjacent sequences: A111072 A111073 A111074 * A111076 A111077 A111078
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KEYWORD
| nonn
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AUTHOR
| Leroy Quet Oct 10 2005
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Emeric Deutsch (deutsch(AT)duke.poly.edu), Paul D. Hanna (pauldhanna(AT)juno.com) and Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 11 2005
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