login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A130095 Inverse Möbius transform of odd-indexed Fibonacci numbers. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A089406

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 12 19:31 EDT 2021. Contains 344960 sequences. (Running on oeis4.)