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A004695 a(n) = floor(Fibonacci(n)/2). 21
0, 0, 0, 1, 1, 2, 4, 6, 10, 17, 27, 44, 72, 116, 188, 305, 493, 798, 1292, 2090, 3382, 5473, 8855, 14328, 23184, 37512, 60696, 98209, 158905, 257114, 416020, 673134, 1089154, 1762289, 2851443, 4613732, 7465176, 12078908, 19544084, 31622993, 51167077, 82790070, 133957148 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Column sums of:

1 1 2 3 5  8 13 21 34 55...

      1 1  2  3  5  8 13...

              1  1  2  3...

                       1...

---------------------------

1 1 2 4 6 10 17 27 44 72...

This sequence counts partially ordered partitions of (n-3) into parts no greater than 3, where the position of the 1's and 2's is important. Alternatively, the position of the 3's is unimportant. (see example below). - David Neil McGrath, Apr 26 2015

Also the matching and vertex cover number of the (n-2)-Fibonacci cube graph. - Eric W. Weisstein, Sep 06 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

H. Matsui et al., Problem B-1019, Fibonacci Quarterly, Vol. 45, Number 2; 2007; p. 182. [A related sequence.]

Eric Weisstein's World of Mathematics, Fibonacci Cube Graph

Eric Weisstein's World of Mathematics, Matching Number

Eric Weisstein's World of Mathematics, Vertex Cover Number

Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

FORMULA

G.f.: x^3/((1-x^3)*(1-x-x^2)). - Ralf Stephan, Jul 22 2003, corrected by Paul Barry

a(n) = Fibonacci(n)/2 - (1-cos(2Pi*n/3))/3. - Paul Barry, Oct 06 2003

From Paul Barry, Jan 14 2005: (Start)

a(n+2) = Sum_{k=0..floor(n/3)} F(n-3*k).

a(n+2) = Sum_{k=0..n} if(mod(n-k, 3)=0, F(k), 0). (End)

a(n+2) = Sum_{k=0..n} F(k)*(cos(2*Pi*(n-k)/3+Pi/3)/3+sqrt(3)*sin(2*Pi*(n-k)/3+Pi/3)/3+1/3). - Paul Barry, Apr 16 2005

a(n) = a(n-1)+a(n-2)+1 if n mod 3 = 0, else a(n) = a(n-1)+a(n-2). - Gary Detlefs, Dec 05 2010

a(n) = Fibonacci(n-2)+floor(Fibonacci(n-3)/2). - Gary Detlefs, Mar 28 2011

a(n) = (A000045(n) - A011655(n))/2.

a(n) = a(n-1)+a(n-2)+a(n-3)-a(n-4)-a(n-5), a(0)=0, a(1)=0, a(2)=0, a(3)=1, a(4)=1. - Carl Najafi, May 06 2014

EXAMPLE

Partial Order of 6 into parts (1,2,3) with position of 3 unimportant. a(9)=17 These are (33),(321=231=213),(312=132=123),(3111=1311=1131=1113),(222),(2211),(2121),(2112),(1212),(1122),(1221),(21111),(12111),(11211),(11121),(11112),(111111). - David Neil McGrath, Apr 26 2015

MAPLE

seq(iquo(fibonacci(n), 2), n=0..36); # Zerinvary Lajos, Apr 20 2008

f:=proc(n) option remember; local t1; if n <= 2 then RETURN(1); fi: if n mod 3 = 1 then t1:=1 else t1:=0; fi: f(n-1)+f(n-2)+t1; end; [seq(f(n), n=1..100)]; # N. J. A. Sloane, May 25 2008

MATHEMATICA

CoefficientList[Series[x^3 / ((1 - x^3) (1 - x - x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

Floor[Fibonacci[Range[0, 50]]/2] (* Harvey P. Dale, Feb 15 2015 *)

LinearRecurrence[{1, 1, 1, -1, -1}, {0, 0, 0, 1, 1}, 50] (* Harvey P. Dale, Feb 15 2015 *)

Floor[Fibonacci[Range[0, 20]]/2] (* Eric W. Weisstein, Sep 06 2017 *)

PROG

(PARI) a(n)=fibonacci(n)\2

(MAGMA) [Floor(Fibonacci(n)/2): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

CROSSREFS

Cf. A036605, A027976, A081410.

Sequence in context: A107742 A228779 A158510 * A014216 A192683 A079961

Adjacent sequences:  A004692 A004693 A004694 * A004696 A004697 A004698

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 11 1996

STATUS

approved

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Last modified July 20 22:20 EDT 2019. Contains 325189 sequences. (Running on oeis4.)