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 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 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 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

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