login
A034744
Dirichlet convolution of Fibonacci numbers with themselves.
1
1, 2, 4, 7, 10, 20, 26, 48, 72, 120, 178, 316, 466, 780, 1240, 2025, 3194, 5268, 8362, 13670, 21944, 35600, 57314, 93156, 150075, 243252, 392972, 636454, 1028458, 1665600, 2692538, 4358718, 7049512, 11408968, 18455060, 29866716, 48315634, 78184700, 126492904
OFFSET
1,2
LINKS
FORMULA
a(n) = sum(Fibonacci(d)*Fibonacci(n/d), d|n). [Emanuele Munarini, Feb 14 2014]
G.f.: Sum_{k>=1} Fibonacci(k) * x^k/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Jul 24 2019
a(n) ~ 2 * ((1+sqrt(5))/2)^n / sqrt(5). - Vaclav Kotesovec, Sep 11 2019
MATHEMATICA
Table[Sum[Fibonacci[d] Fibonacci[n/d], {d, Divisors[n]}], {n, 1, 100}] (* Emanuele Munarini, Feb 14 2014 *)
PROG
(Maxima) a(n) := lsum(fib(d)*fib(n/d), d, listify(divisors(n)));
makelist(a(n), n, 1, 40); /* Emanuele Munarini, Feb 14 2014 */
(PARI) a(n) = sumdiv(n, d, fibonacci(d)*fibonacci(n/d)); \\ Michel Marcus, Feb 14 2014
(Magma) [&+[Fibonacci(d)*Fibonacci(n div d): d in Divisors(n)]: n in [1..40]]; // Bruno Berselli, Feb 11 2014
CROSSREFS
Cf. A000045.
Sequence in context: A353055 A173730 A036685 * A219748 A121352 A134126
KEYWORD
nonn
STATUS
approved