OFFSET
1,2
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
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
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