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Expansion of 1/(1-4*x+x^3).
5

%I #24 Aug 04 2023 01:43:22

%S 1,4,16,63,248,976,3841,15116,59488,234111,921328,3625824,14269185,

%T 56155412,220995824,869714111,3422701032,13469808304,53009519105,

%U 208615375388,820991693248,3230957253887,12715213640160,50039862867392

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

%C A transform of A000302 under the mapping g(x) ->(1/(1+x^3)) * g(x/(1+x^3)).

%H G. C. Greubel, <a href="/A099503/b099503.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-1).

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*4^(n-3*k).

%t CoefficientList[Series[1/(1-4x+x^3),{x,0,30}],x] (* _Harvey P. Dale_, Apr 01 2011 *)

%t LinearRecurrence[{4,0,-1}, {1,4,16}, 30] (* _G. C. Greubel_, Aug 03 2023 *)

%o (Magma) [n le 3 select 4^(n-1) else 4*Self(n-1) -Self(n-3): n in [1..30]]; // _G. C. Greubel_, Aug 03 2023

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A099503

%o if (n<3): return 4^n

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

%o [a(n) for n in range(31)] # _G. C. Greubel_, Aug 03 2023

%Y Cf. A000071, A000302, A076264, A099504.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 20 2004