A130095 - OEIS

%I #11 Dec 11 2019 06:58:45

%S 1,3,6,16,35,97,234,626,1603,4218,10947,28767,75026,196654,514269,

%T 1346895,3524579,9229159,24157818,63250217,165580380,433505386,

%U 1134903171,2971244450,7778742084,20365086102,53316292776,139584059112,365435296163,956722544582

%N Inverse Möbius transform of odd-indexed Fibonacci numbers.

%C Original name was: A051731 * A007436.

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

%F From _Peter Bala_, Mar 26 2015: (Start)

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

%F 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)).

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

%F 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)

%e The divisors of 6 are 1, 2, 3 and 6. Hence

%e a(6) = Fibonacci(1) + Fibonacci(3) + Fibonacci(5) + Fibonacci(11) = 97.

%p #A130095

%p with(combinat): with(numtheory):

%p f := n -> fibonacci(2*n-1):

%p g := proc (n) local div; div := divisors(n):

%p add(f(div[j]), j = 1 .. tau(n)) end proc:

%p seq(g(n), n = 1 .. 30); # _Peter Bala_, Mar 26 2015

%Y Cf. A001519, A007435, A051731.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, May 06 2007

%E Incorrect original name removed and terms a(11) - a(30) added by _Peter Bala_, Mar 26 2015