%I #26 Aug 31 2024 08:34:28

%S 1,6,25,111,488,2149,9461,41654,183389,807403,3554736,15650361,

%T 68903513,303360038,1335596817,5880203831,25888648920,113979406525,

%U 501814720109,2209329044566,9726966211957,42824708216851,188543436246752,830096195208753,3654646945111665

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

%H G. C. Greubel, <a href="/A100296/b100296.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+2*x-x^2)/(1-4*x-2*x^2+x^3). - _Colin Barker_, May 25 2013

%e a(8) = 4*a(7) + 2*a(6) - a(5) = 41654 = 4*9461 + 2*2149 - 488.

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

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Feb 06 2023

%t LinearRecurrence[{4,2,-1}, {1,6,25}, 40] (* _G. C. Greubel_, Feb 05 2023 *)

%o (Magma) I:=[1,6,25]; [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 (1,1,6)[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. A100295, 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