OFFSET
0,5
COMMENTS
Also the independence number of the n-Fibonacci cube graph. - Eric W. Weisstein, Sep 06 2017
Also the edge cover number of the (n-2)-Fibonacci cube graph. - Eric W. Weisstein, Dec 26 2017
Also the calque covering number of the (n-2)-Fibonacci cube graph. - Eric W. Weisstein, Apr 20 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..280
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Edge Cover Number
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
Eric Weisstein's World of Mathematics, Independence Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).
FORMULA
a(n) = ceiling(Fibonacci(n)/2). - Mircea Merca, Jan 04 2010
a(n) = a(n-1) +a(n-2) +a(n-3) -a(n-4) -a(n-5) - Joerg Arndt, Apr 24 2011
G.f.: x/(1 - x*(1-x^4)/(1 - x^2*(1-x^5)/(1 - x^3*(1-x^6)/(1 - x^4*(1-x^7)/(1 - x^5*(1-x^8)/(1 - x^6*(1-x^9)/(1 - x^7*(1-x^10)/(1 - x^8*(1-x^11)/(1 - ...))))))))), (continued fraction) - Paul D. Hanna, Jul 08 2013
G.f.: x*(1 - x^2 - x^3) / ((1-x^3)*(1 - x - x^2)). [Paul D. Hanna, Jul 18 2013, from Joerg Arndt's formula]
For n > 1, if n == 0 (mod 3) then a(n) = a(n-1) + a(n-2) - 1; otherwise a(n) = a(n-1) + a(n-2). - Franklin T. Adams-Watters, Jun 11 2018
MAPLE
with(combinat, fibonacci): seq(ceil(fibonacci(n)/2), n=0..33) # Mircea Merca, Jan 04 2010
MATHEMATICA
Table[Fibonacci[n] - Floor[Fibonacci[n]/2], {n, 0, 40}] (* Harvey P. Dale, Jun 09 2013 *)
(* Start from Eric W. Weisstein, Sep 06 2017 *)
Table[Ceiling[Fibonacci[n]/2], {n, 0, 20}]
Ceiling[Fibonacci[Range[0, 20]]/2]
LinearRecurrence[{1, 1, 1, -1, -1}, {1, 2, 3, 4, 7}, 20]
CoefficientList[Series[(1 + x - 2 x^3 - x^4)/(1 - x - x^2 - x^3 + x^4 + x^5), {x, 0, 20}], x]
(* End *)
PROG
(Magma) [Fibonacci(n) - Floor(Fibonacci(n)/2): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011
(PARI) /* Continued Fraction: */
{a(n)=my(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+4)) *CF +x*O(x^n) )); polcoeff(x*CF, n)} \\ Paul D. Hanna, Jul 08 2013
(PARI) {a(n)=polcoeff( x*(1 - x^2 - x^3) / ((1-x^3)*(1 - x - x^2 +x*O(x^n))), n)} \\ Paul D. Hanna, Jul 18 2013
(PARI) a(n)=(fibonacci(n)+1)\2 \\ Charles R Greathouse IV, Jun 11 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Nov 22 2010
EXTENSIONS
Name simplified using Mircea Merca's formula by Eric W. Weisstein, Sep 06 2017
STATUS
approved