A130095 Inverse Möbius transform of odd-indexed Fibonacci numbers.

1, 3, 6, 16, 35, 97, 234, 626, 1603, 4218, 10947, 28767, 75026, 196654, 514269, 1346895, 3524579, 9229159, 24157818, 63250217, 165580380, 433505386, 1134903171, 2971244450, 7778742084, 20365086102, 53316292776, 139584059112, 365435296163, 956722544582

Offset

1,2

Comments

Original name was: A051731 * A007436.

Conjecture: a(n)/a(n-1) tends to phi^2.

Links

Table of n, a(n) for n=1..30.

Formula

From Peter Bala, Mar 26 2015: (Start)

a(n) = sum {d | n} Fibonacci(2*d - 1).

O.g.f. Sum_{n >= 1} Fibonacci(2*n - 1)*x^n/(1 - x^n) = Sum_{n >= 1} x^n*(1 - x^n)/(1 - 3*x^n + x^(2*n)).

Sum_{n >= 1} a(n)*x^(2*n) = Sum_{n >= 1} x^n/( 1/(x^n - 1/x^n) - (x^n - 1/x^n) ).

For p prime, a(p) == k (mod p) where k = 3 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 0 if p = 5. (End)

Example

The divisors of 6 are 1, 2, 3 and 6. Hence
a(6) = Fibonacci(1) + Fibonacci(3) + Fibonacci(5) + Fibonacci(11) = 97.

Maple

#A130095
with(combinat): with(numtheory):
f := n -> fibonacci(2*n-1):
g := proc (n) local div; div := divisors(n):
add(f(div[j]), j = 1 .. tau(n)) end proc:
seq(g(n), n = 1 .. 30); # Peter Bala, Mar 26 2015

Crossrefs

Cf. A001519, A007435, A051731.

Sequence in context: A068590 A327736 A319752 * A293993 A072824 A360229

Adjacent sequences: A130092 A130093 A130094 * A130096 A130097 A130098

Keyword

nonn,easy

Author

Gary W. Adamson, May 06 2007

Extensions

Incorrect original name removed and terms a(11) - a(30) added by Peter Bala, Mar 26 2015

Status

approved