A123871 - OEIS

%I #30 Sep 08 2022 08:45:28

%S 1,5,25,120,580,2800,13520,65280,315200,1521920,7348480,35481600,

%T 171320320,827207680,3994112000,19285278720,93117562880,449611366400,

%U 2170915717120,10482108334080,50612096204800,244376818155520,1179955657441280,5697329902387200

%N Expansion of g.f.: (1+x+x^2)/(1-4*x-4*x^2).

%H Vincenzo Librandi, <a href="/A123871/b123871.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0112281">Words restricted by 3-letter ...</a>, Annals of Combinatorics 7 (2003), 1-14. arXiv:math.CO/0112281

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.pdf">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).

%F a(n) = 4*a(n-1) + 4*a(n-2) for n>2. - _Philippe Deléham_, Sep 19 2009

%p seq(coeff(series((1+x+x^2)/(1-4*x-4*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Aug 08 2019

%t CoefficientList[Series[(1+x+x^2)/(1-4*x-4*x^2),{x,0,30}],x] (* _Vincenzo Librandi_, Jun 27 2012 *)

%t LinearRecurrence[{4,4},{1,5,25},30] (* _Harvey P. Dale_, Mar 25 2022 *)

%o (Magma) I:=[1, 5, 25]; [n le 3 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 27 2012

%o (PARI) my(x='x+O('x^30)); Vec((1+x+x^2)/(1-4*x-4*x^2)) \\ _G. C. Greubel_, Aug 08 2019

%o (Sage)

%o def A123871_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1+x+x^2)/(1-4*x-4*x^2) ).list()

%o A123871_list(30) # _G. C. Greubel_, Aug 08 2019

%o (GAP) a:=[1,5,25];; for n in [4..30] do a[n]:=4*a[n-1]+4*a[n-2]; od; a; # _G. C. Greubel_, Aug 08 2019

%Y Column 5 in A265584.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 20 2006