A349451 Dirichlet inverse of Fibonacci numbers, when started from A000045(1): 1, 1, 2, 3, 5, 8, 13, 21, ...

1, -1, -2, -2, -5, -4, -13, -16, -30, -45, -89, -122, -233, -351, -590, -944, -1597, -2496, -4181, -6640, -10894, -17533, -28657, -46000, -75000, -120927, -196290, -317018, -514229, -830580, -1346269, -2176288, -3524222, -5699693, -9227335, -14924550, -24157817, -39079807, -63245054, -102320320, -165580141, -267890844

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001

Crossrefs

Cf. A000045.

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

Programs

Mathematica

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

PARI

memoA349451 = Map();
A349451(n) = if(1==n,1,my(v); if(mapisdefined(memoA349451,n,&v), v, v = -sumdiv(n,d,if(dA349451(d),0)); mapput(memoA349451,n,v); (v)));

Formula

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

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