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

%I #20 Aug 31 2024 08:33:34

%S 1,3,14,61,269,1184,5213,22951,101046,444873,1958633,8623232,37965321,

%T 167149115,735903870,3239948389,14264452181,62801801632,276496162501,

%U 1217323801087,5359485727718,23596094350545,103886025056529,457376803199488,2013683168560465

%N Expansion of g.f. x*(1-x)/(1-4*x-2*x^2+x^3).

%H G. C. Greubel, <a href="/A100295/b100295.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,-1).

%F a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3).

%F G.f.: x*(1-x)/(1-4*x-2*x^2+x^3). - _Colin Barker_, May 25 2013

%p a:= n-> (<<3|2|1>, <2|1|0>, <1|0|0>>^n)[1,3]:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, May 25 2013

%t LinearRecurrence[{4,2,-1}, {1,3,14}, 40] (* _G. C. Greubel_, Feb 05 2023 *)

%o (Magma) I:=[1,3,14]; [n le 3 select I[n] else 4*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // _G. C. Greubel_, Feb 05 2023

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A100296

%o if (n<3): return (0,1,3)[n]

%o else: return 4*a(n-1) + 2*a(n-2) - a(n-3)

%o [a(n) for n in range(1,41)] # _G. C. Greubel_, Feb 05 2023

%Y Cf. A100296, A100297.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Nov 11 2004

%E More terms from _Colin Barker_, May 25 2013

%E New name using g.f. from _Joerg Arndt_, Aug 31 2024