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A071100 Expansion of (5 + 3*x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.

%I A071100 #35 Sep 03 2021 14:30:18
%S A071100 5,13,37,109,313,905,2617,7561,21853,63157,182525,527509,1524529,
%T A071100 4405969,12733489,36800465,106355317,307372573,888323221,2567301757,
%U A071100 7419639785,21443156953,61971873769,179102039257,517614500173,1495933669445,4323328543981
%N A071100 Expansion of (5 + 3*x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.
%C A071100 Number of tilings of the 0-mod-4 pillow of order n is a perfect square times a(n). [Propp, 1999, p. 271]
%D A071100 J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 12).
%H A071100 A.H.M. Smeets, <a href="/A071100/b071100.txt">Table of n, a(n) for n = 0..2169</a>
%H A071100 J. Propp, <a href="http://faculty.uml.edu/jpropp/update.pdf">Updated article</a>
%H A071100 J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H A071100 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F A071100 G.f.: (5 + 3*x + x^2 -x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).
%F A071100 a(-n) = a(-5 + n). a(-1) = a(-2) = 1. a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4). - _Michael Somos_, Dec 15 2011
%F A071100 A112835(2*n + 2) = a(n).
%F A071100 Lim_{n -> inf} a(n)/a(n-1) = A318605. - _A.H.M. Smeets_, Sep 12 2018
%e A071100 G.f. = 5 + 13*x + 37*x^2 + 109*x^3 + 313*x^4 + 905*x^5 + 2617*x^6 + 7561*x^7 + ...
%p A071100 seq(coeff(series((5+3*x+x^2-x^3)/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Sep 12 2018
%t A071100 CoefficientList[Series[(5 + 3*x + x^2 -x^3)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4), {x, 0, 50}], x] (* _Stefano Spezia_, Sep 12 2018 *)
%t A071100 LinearRecurrence[{2,2,2,-1},{5,13,37,109},30] (* _Harvey P. Dale_, Sep 03 2021 *)
%o A071100 (PARI) {a(n) = my(m = n+2); if( m < 0, m = -1 - m); polcoeff( (1 - x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + x * O(x^m), m)}; /* _Michael Somos_, Dec 15 2011 */
%o A071100 (PARI) x='x+O('x^33); Vec((5+3*x+x^2-x^3)/(1-2*x-2*x^2-2*x^3+x^4)) \\ _Altug Alkan_, Sep 12 2018
%o A071100 (GAP) a:=[5,13,37,109];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # _Muniru A Asiru_, Sep 12 2018
%Y A071100 Cf. A112835.
%K A071100 nonn,easy
%O A071100 0,1
%A A071100 _N. J. A. Sloane_, May 28 2002