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A052941 Expansion of (1-x)/(1 - 4*x + x^2 + x^3).

%I A052941 #45 Jul 02 2025 16:01:58
%S A052941 1,3,11,40,146,533,1946,7105,25941,94713,345806,1262570,4609761,
%T A052941 16830668,61450341,224360935,819162731,2990839648,10919834926,
%U A052941 39869337325,145566674726,531477526653,1940474094561,7084852176865
%N A052941 Expansion of (1-x)/(1 - 4*x + x^2 + x^3).
%C A052941 a(n) = term (3,1) in M^n, M = the 3 X 3 matrix [1,1,2; 1,2,1; 1,1,1]. - _Gary W. Adamson_, Mar 12 2009
%H A052941 Vincenzo Librandi, <a href="/A052941/b052941.txt">Table of n, a(n) for n = 0..1000</a>
%H A052941 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A052941 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=932">Encyclopedia of Combinatorial Structures 932</a>
%H A052941 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-1).
%F A052941 G.f.: (1-x)/(1 - 4*x + x^2 + x^3).
%F A052941 a(n) = 4*a(n-1) - a(n-2) - a(n-3).
%F A052941 a(n) = Sum_{alpha=RootOf(1-4*z+z^2+z^3)} (3-alpha^2)*alpha^(-1-n)/13.
%F A052941 a(n) = (b(n+2) - b(n+1) + b(n))/13, where b(n) = A356463(n). - _Ding Hao_, Aug 08 2022
%p A052941 spec:= [S,{S=Sequence(Union(Z,Z,Prod(Union(Sequence(Z),Z),Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p A052941 seq(coeff(series((1-x)/(1-4*x+x^2+x^3), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 18 2019
%t A052941 LinearRecurrence[{4,-1,-1},{1,3,11},30] (* _Vincenzo Librandi_, Jun 22 2012 *)
%o A052941 (Magma) I:=[1, 3, 11]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 22 2012
%o A052941 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-4*x+x^2+x^3)) \\ _Altug Alkan_, Sep 21 2018
%o A052941 (Sage)
%o A052941 def A052941_list(prec):
%o A052941     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052941     return P((1-x)/(1-4*x+x^2+x^3)).list()
%o A052941 A052941_list(30) # _G. C. Greubel_, Oct 18 2019
%o A052941 (GAP) a:=[1,3,11];; for n in [4..30] do a[n]:=4*a[n-1]-a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Oct 18 2019
%K A052941 easy,nonn
%O A052941 0,2
%A A052941 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052941 More terms from _James Sellers_, Jun 06 2000