cp's OEIS Frontend

A176737 Expansion of 1 / (1-4*x^2-3*x^3). (4,3)-Padovan sequence.

%I A176737 #22 Dec 26 2017 11:08:25
%S A176737 1,0,4,3,16,24,73,144,364,795,1888,4272,9937,22752,52564,120819,
%T A176737 278512,640968,1476505,3399408,7828924,18027147,41513920,95595360,
%U A176737 220137121,506923200,1167334564,2688104163,6190107856,14254420344,32824743913,75588004944,174062236684
%N A176737 Expansion of 1 / (1-4*x^2-3*x^3). (4,3)-Padovan sequence.
%C A176737 See A000931 (Padovan), and the W. Lang link given there.
%H A176737 Colin Barker, <a href="/A176737/b176737.txt">Table of n, a(n) for n = 0..1000</a>
%H A176737 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,3).
%F A176737 O.g.f.: 1/((1-x-3*x^2)*(1+x)) = (2-3*x)/(1-x-3*x^2) -1/(1+x).
%F A176737 a(n) = 2*b(n) - 3*b(n-1) - (-1)^n, n>=0, with b(n):=A006130(n) ((1,3)-Fibonacci), b(-1):=0.
%F A176737 From _Wolfdieter Lang_, Aug 26 2010: (Start)
%F A176737 a(n) = a(n-1) + 3*a(n-2) + (-1)^n, n>=2, a(0)=1, a(1)=0.
%F A176737 Due to the identity for the o.g.f. A(x): A(x)= x*(1 + 3*x)*A(x) + 1/(1+x).
%F A176737 (This recurrence was observed by _Gary Detlefs_ in an Aug 24 2010 email to the author.)
%F A176737 (End)
%F A176737 a(n) = 4*a(n-2) + 3*a(n-3) for n>2. - _Harvey P. Dale_, Jan 21 2013
%F A176737 a(n) = (-1)^(n+1)*A140165(n+2)-(-1)^n. - _R. J. Mathar_, Apr 22 2013
%F A176737 a(n) = ((-1)^(1+n) + (2^(-n)*((-2+sqrt(13))*(1+sqrt(13))^n + (1-sqrt(13))^n*(2+sqrt(13)))) / sqrt(13)). - _Colin Barker_, Dec 25 2017
%t A176737 CoefficientList[Series[1/(1-4*x^2-3*x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {0,4,3},{1,0,4},40] (* _Harvey P. Dale_, Jan 21 2013 *)
%o A176737 (PARI) Vec(1 / ((1 + x)*(1 - x - 3*x^2)) + O(x^40)) \\ _Colin Barker_, Dec 25 2017
%Y A176737 Cf. A053088 ((3,2)-Padovan).
%K A176737 nonn,easy
%O A176737 0,3
%A A176737 _Wolfdieter Lang_, Jun 26 2010