A034748 Dirichlet convolution of Fibonacci numbers with phi(n).

1, 2, 4, 6, 9, 14, 19, 30, 44, 68, 99, 168, 245, 402, 636, 1026, 1613, 2650, 4199, 6854, 10996, 17820, 28679, 46596, 75065, 121650, 196516, 318250, 514257, 832826, 1346299, 2179374, 3524796, 5704516, 9227571, 14933352, 24157853, 39092386

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000 (first 300 terms from Paolo P. Lava)

Formula

From definition a(n) = Sum_{d|n} F(d)*phi(n/d); also a(n) = Sum_{k=1..n} gcd(F(k), F(k+n)) where F(k) denotes the k-th Fibonacci number. - Benoit Cloitre, May 25 2003

G.f.: Sum_{k>=1} phi(k) * x^k/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Jul 23 2019

a(n) ~ phi^n / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 23 2019

From Richard L. Ollerton, May 06 2021: (Start)

a(n) = Sum_{k=1..n} F(gcd(n,k)).

a(n) = Sum_{k=1..n} F(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Mathematica

Table[Sum[Fibonacci[d] EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 50}] (* Vincenzo Librandi, Aug 19 2018 *)

Prog

(PARI) a(n)=sumdiv(n, d, fibonacci(d)*eulerphi(n/d))
(Magma) [&+[Fibonacci(d)*EulerPhi(n div d): d in Divisors(n)]: n in [1..50]]; // Vincenzo Librandi, Aug 19 2018

Crossrefs

Cf. A000010, A000045, A001622.

Sequence in context: A328863 A218004 A346634 * A069916 A153140 A295341

Adjacent sequences: A034745 A034746 A034747 * A034749 A034750 A034751

Keyword

nonn

Author

Erich Friedman

Status

approved