%I #19 Feb 24 2022 08:58:54

%S 1,-1,-2,-2,-5,-4,-13,-16,-30,-45,-89,-122,-233,-351,-590,-944,-1597,

%T -2496,-4181,-6640,-10894,-17533,-28657,-46000,-75000,-120927,-196290,

%U -317018,-514229,-830580,-1346269,-2176288,-3524222,-5699693,-9227335,-14924550,-24157817,-39079807,-63245054,-102320320,-165580141,-267890844

%N Dirichlet inverse of Fibonacci numbers, when started from A000045(1): 1, 1, 2, 3, 5, 8, 13, 21, ...

%H Antti Karttunen, <a href="/A349451/b349451.txt">Table of n, a(n) for n = 1..1001</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d < n} A000045(n/d) * a(d).

%F G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Fibonacci(k) * A(x^k). - _Ilya Gutkovskiy_, Feb 23 2022

%t a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * Fibonacci[n/#] &, # < n &]; Array[a, 42] (* _Amiram Eldar_, Nov 22 2021 *)

%o (PARI)

%o memoA349451 = Map();

%o A349451(n) = if(1==n,1,my(v); if(mapisdefined(memoA349451,n,&v), v, v = -sumdiv(n,d,if(d<n,fibonacci(n/d)*A349451(d),0)); mapput(memoA349451,n,v); (v)));

%Y Cf. A000045.

%Y Cf. also A349449, A349450, A349452, A349453, A349566.

%K sign

%O 1,3

%A _Antti Karttunen_, Nov 22 2021