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%I #26 Sep 13 2023 12:20:11
%S 8,1,41,142,933,4741,26660,143697,788453,4293286,23454801,127953981,
%T 698467368,3811712633,20803963753,113540081302,619672701957,
%U 3381980484909,18457878595412,100737602247769,549796303339413
%N Number of independent sets of nodes in P_4 X C_n (n > 2).
%H Vincenzo Librandi, <a href="/A050401/b050401.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,20,27,-14,-25,4,5,-1).
%F a(n) = a(n-1) + 20*a(n-2) + 27*a(n-3) - 14*a(n-4) - 25*a(n-5) + 4*a(n-6) + 5*a(n-7) - a(n-8).
%F G.f.: (8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)). - _Colin Barker_, Aug 31 2012
%p seq(coeff(series((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/( (1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)), x, n+1), x, n), n = 0 ..30); # _G. C. Greubel_, Oct 30 2019
%t CoefficientList[Series[(8-7*x-120*x^2-135*x^3+56*x^4+75*x^5-8*x^6-5*x^7) /( (1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5)), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 11 2017 *)
%o (Magma) I:=[8,1,41,142,933,4741,26660,143697]; [n le 8 select I[n] else Self(n-1)+20*Self(n-2)+27*Self(n-3)-14*Self(n-4)- 25*Self(n-5)+4*Self(n-6)+5*Self(n-7)-Self(n-8): n in [1..30]]; // _Vincenzo Librandi_, May 11 2017
%o (PARI) my(x='x+O('x^30)); Vec((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5))) \\ _G. C. Greubel_, Oct 30 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*( 1-4*x-9*x^2+5*x^3+4*x^4-x^5)) )); // _G. C. Greubel_, Oct 30 2019
%o (Sage)
%o def A050401_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((8 -7*x -120*x^2 -135*x^3 +56*x^4 +75*x^5 -8*x^6 -5*x^7)/((1+x)*(1+2*x-x^2)*(1-4*x-9*x^2+5*x^3+4*x^4-x^5))).list()
%o A050401_list(30) # _G. C. Greubel_, Oct 30 2019
%o (GAP) a:=[8,1,41,142,933,4741,26660,143697];; for n in [9..30] do a[n]:= a[n-1]+20*a[n-2]+27*a[n-3]-14*a[n-4]-25*a[n-5]+4*a[n-6]+5*a[n-7]-a[n-8]; od; a; # _G. C. Greubel_, Oct 30 2019
%Y Column 4 of A286513.
%K easy,nonn
%O 0,1
%A _Stephen G Penrice_, Dec 21 1999
%E More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999
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