|
|
A007435
|
|
Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...
(Formerly M0631)
|
|
13
|
|
|
1, 2, 3, 5, 6, 12, 14, 26, 37, 62, 90, 159, 234, 392, 618, 1013, 1598, 2630, 4182, 6830, 10962, 17802, 28658, 46548, 75031, 121628, 196455, 318206, 514230, 832722, 1346270, 2179322, 3524670, 5704486, 9227484, 14933129, 24157818, 39092352, 63246222, 102341006
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
For p prime, a(p) == k (mod p) where k = 0 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 1 if p = 5. - Michael Somos, Apr 15 2012
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
G.f.: Sum_{k>0} Fibonacci(k)*x^k/(1-x^k) = Sum_{k>0} x^k/(1-x^k-x^(2*k)). - Vladeta Jovovic, Dec 17 2002
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(Fibonacci(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 20 2018
a(n) ~ 5^(-1/2) * phi^n, where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2018
|
|
EXAMPLE
|
x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 12*x^6 + 14*x^7 + 26*x^8 + 37*x^9 + 62*x^10 + ...
|
|
MATHEMATICA
|
Table[Plus @@ Map[Function[d, Fibonacci[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
|
|
PROG
|
(PARI) {a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* Michael Somos, Apr 15 2012 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|