A038990 - OEIS

%I #20 Jul 16 2017 05:55:09

%S 1,2,5,14,37,98,257,674,1765,4622,12101,31682,82945,217154,568517,

%T 1488398,3896677,10201634,26708225,69923042,183060901,479259662,

%U 1254718085,3284894594,8599965697,22515002498,58945041797,154320122894,404015326885,1057725857762

%N Expansion of (1-x-x^2+2*x^3) / ((1-x)*(1+x)*(1-3*x+x^2)).

%H Colin Barker, <a href="/A038990/b038990.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).

%F a(n) = -1/2-(-1)^n/10+4*(2*A001906(n+1)-3*A001906(n))/5. - _R. J. Mathar_, Mar 31 2011

%F a(0)=1, a(1)=2, a(2)=5, a(3)=14, a(n)=3*a(n-1)-3*a(n-3)+a(n-4). - _Harvey P. Dale_, Feb 17 2012

%F a(n) = (-1)*(2^(-1-n)*((-2)^n + 5*2^n - 8*(3-sqrt(5))^n - 8*(3+sqrt(5))^n)) / 5. - _Colin Barker_, Jul 16 2017

%p A001906 := proc(n) combinat[fibonacci](2*n) ; end proc:

%p A038990 := proc(n) -1/2-(-1)^n/10+4*(2*A001906(n+1)-3*A001906(n))/5 ; end proc: # _R. J. Mathar_, Mar 31 2011

%t CoefficientList[Series[(1-x-x^2+2x^3)/((1-x)(1+x)(1-3x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{3,0,-3,1},{1,2,5,14},30] (* _Harvey P. Dale_, Feb 17 2012 *)

%o (PARI) Vec((1-x-x^2+2*x^3)/((1-x)*(1+x)*(1-3*x+x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%K nonn,easy

%O 0,2

%A _Colin Mallows_