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A004695 a(n) = floor(Fibonacci(n)/2).

%I A004695 #85 Feb 16 2025 08:32:28
%S A004695 0,0,0,1,1,2,4,6,10,17,27,44,72,116,188,305,493,798,1292,2090,3382,
%T A004695 5473,8855,14328,23184,37512,60696,98209,158905,257114,416020,673134,
%U A004695 1089154,1762289,2851443,4613732,7465176,12078908,19544084,31622993,51167077,82790070,133957148
%N A004695 a(n) = floor(Fibonacci(n)/2).
%C A004695 Column sums of:
%C A004695 1 1 2 3 5  8 13 21 34 55...
%C A004695       1 1  2  3  5  8 13...
%C A004695               1  1  2  3...
%C A004695                        1...
%C A004695 ---------------------------
%C A004695 1 1 2 4 6 10 17 27 44 72...
%C A004695 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
%C A004695 Also the matching and vertex cover number of the (n-2)-Fibonacci cube graph. - _Eric W. Weisstein_, Sep 06 2017
%H A004695 Vincenzo Librandi, <a href="/A004695/b004695.txt">Table of n, a(n) for n = 0..1000</a>
%H A004695 H. Matsui et al., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/elementary45-2.pdf">Problem B-1019</a>, Fibonacci Quarterly, Vol. 45, Number 2; 2007; p. 182. [A related sequence.]
%H A004695 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciCubeGraph.html">Fibonacci Cube Graph</a>
%H A004695 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>
%H A004695 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCoverNumber.html">Vertex Cover Number</a>
%H A004695 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).
%F A004695 G.f.: x^3/((1-x^3)*(1-x-x^2)). - _Ralf Stephan_, Jul 22 2003, corrected by _Paul Barry_
%F A004695 a(n) = Fibonacci(n)/2 - (1-cos(2Pi*n/3))/3. - _Paul Barry_, Oct 06 2003
%F A004695 From _Paul Barry_, Jan 14 2005: (Start)
%F A004695 a(n+2) = Sum_{k=0..floor(n/3)} F(n-3*k).
%F A004695 a(n+2) = Sum_{k=0..n} if(mod(n-k, 3)=0, F(k), 0). (End)
%F A004695 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
%F A004695 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
%F A004695 a(n) = Fibonacci(n-2)+floor(Fibonacci(n-3)/2). - _Gary Detlefs_, Mar 28 2011
%F A004695 a(n) = (A000045(n) - A011655(n))/2.
%F A004695 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
%e A004695 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
%p A004695 seq(iquo(fibonacci(n),2),n=0..36); # _Zerinvary Lajos_, Apr 20 2008
%p A004695 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
%t A004695 CoefficientList[Series[x^3 / ((1 - x^3) (1 - x - x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 08 2013 *)
%t A004695 Floor[Fibonacci[Range[0, 50]]/2] (* _Harvey P. Dale_, Feb 15 2015 *)
%t A004695 LinearRecurrence[{1, 1, 1, -1, -1}, {0, 0, 0, 1, 1}, 50] (* _Harvey P. Dale_, Feb 15 2015 *)
%t A004695 Floor[Fibonacci[Range[0, 20]]/2] (* _Eric W. Weisstein_, Sep 06 2017 *)
%o A004695 (PARI) a(n)=fibonacci(n)\2
%o A004695 (Magma) [Floor(Fibonacci(n)/2): n in [0..60]]; // _Vincenzo Librandi_, Apr 23 2011
%Y A004695 Cf. A036605, A027976, A081410.
%K A004695 nonn,easy
%O A004695 0,6
%A A004695 _N. J. A. Sloane_, Dec 11 1996