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

Author

Roger L. Bagula, Nov 22 2010

Keywords

• nonn
• easy

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).

Crossrefs

Column m=3 of A185646.

Programs

Magma

[Fibonacci(n) - Floor(Fibonacci(n)/2): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

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 *)

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

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

Extensions

Name simplified using Mircea Merca's formula by Eric W. Weisstein, Sep 06 2017