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%I #46 Apr 15 2024 11:16:22
%S 1,1,2,4,9,19,40,85,180,381,807,1709,3619,7664,16230,34370,72785,
%T 154136,326412,691239,1463829,3099934,6564695,13901980,29440065,
%U 62344891,132027067,279592219,592089264,1253860704,2655286560
%N Expansion of 1 + x/(1 - 2*x - x^3 + x^4).
%C Let A(r,n) count the total number of ordered arrangements of an n+r tiling of r red squares and white tiles of total length n, where the individual tile lengths can range from 1 to n. A(r,0) corresponds to a tiling of r red squares only. Let A_1(r,n) = Sum_{j=0..n} A(r,j). Then the expansion of 1/(1-2*x-x^3+x^4) = A_1(0,n) + A_1(1,n-3) + A_1(2, n-6) + ..., which generates a(n) without the initial 1. - _Gregory L. Simay_, May 24 2018
%H G. C. Greubel, <a href="/A052908/b052908.txt">Table of n, a(n) for n = 0..1000</a>
%H Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, Mark Shattuck, <a href="https://www.emis.de/journals/JIS/VOL22/Shattuck/shattuck13.html">Some New Restricted n-Color Composition Functions</a>, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=888">Encyclopedia of Combinatorial Structures 888</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1,-1).
%F G.f.: (1-x)*(1-x^3)/(1-2*x-x^3+x^4).
%F a(n) = 2*a(n-1) + a(n-3) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=4, a(4)=9.
%F a(n) = Sum_{alpha=RootOf(1 -2*z -z^3 +z^4)} (1/643)*(168 - 74*alpha + 53*alpha^2 - 93*alpha^3)*alpha^(-1-n).
%F a(n) = A052903(n)-A052903(n-1). - _R. J. Mathar_, Apr 15 2024
%p spec := [S,{S=Sequence(Prod(Sequence(Z),Sequence(Prod(Z,Z,Z)),Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t CoefficientList[Series[(1-x)(1-x^3)/(1 -2x -x^3 +x^4), {x, 0, 30}], x] (* _Michael De Vlieger_, Jun 14 2018 *)
%t LinearRecurrence[{2,0,1,-1}, {1,1,2,4,9}, 30}] (* _G. C. Greubel_, Oct 14 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-x)*(1-x^3)/(1-2*x-x^3+x^4)) \\ _G. C. Greubel_, Oct 14 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)*(1-x^3)/(1-2*x-x^3+x^4) )); // _G. C. Greubel_, Oct 14 2019
%o (Sage)
%o def A052908_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-x)*(1-x^3)/(1-2*x-x^3+x^4) ).list()
%o A052908_list(30) # _G. C. Greubel_, Oct 14 2019
%o (GAP) a:=[1,2,4,9];; for n in [5..30] do a[n]:=2*a[n-1]+a[n-3]-a[n-4]; od; Concatenation([1], a); # _G. C. Greubel_, Oct 14 2019
%K easy,nonn,changed
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000
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