A173173 a(n) = ceiling(Fibonacci(n)/2).

0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 45, 72, 117, 189, 305, 494, 799, 1292, 2091, 3383, 5473, 8856, 14329, 23184, 37513, 60697, 98209, 158906, 257115, 416020, 673135, 1089155, 1762289, 2851444, 4613733, 7465176, 12078909, 19544085, 31622993, 51167078, 82790071

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]

a(n) = A061347(n)/6 +1/3 +A000045(n)/2. - R. J. Mathar, Jul 19 2013

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

Column m=3 of A185646.

Sequence in context: A222026 A293672 A050193 * A303025 A346020 A192669

Adjacent sequences: A173170 A173171 A173172 * A173174 A173175 A173176

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