cp's OEIS Frontend

A099503 Expansion of 1/(1-4*x+x^3).

%I A099503 #24 Aug 04 2023 01:43:22
%S A099503 1,4,16,63,248,976,3841,15116,59488,234111,921328,3625824,14269185,
%T A099503 56155412,220995824,869714111,3422701032,13469808304,53009519105,
%U A099503 208615375388,820991693248,3230957253887,12715213640160,50039862867392
%N A099503 Expansion of 1/(1-4*x+x^3).
%C A099503 A transform of A000302 under the mapping g(x) ->(1/(1+x^3)) * g(x/(1+x^3)).
%H A099503 G. C. Greubel, <a href="/A099503/b099503.txt">Table of n, a(n) for n = 0..1000</a>
%H A099503 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-1).
%F A099503 a(n) = 4*a(n-1) - a(n-3).
%F A099503 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k*4^(n-3*k).
%t A099503 CoefficientList[Series[1/(1-4x+x^3),{x,0,30}],x]  (* _Harvey P. Dale_, Apr 01 2011 *)
%t A099503 LinearRecurrence[{4,0,-1}, {1,4,16}, 30] (* _G. C. Greubel_, Aug 03 2023 *)
%o A099503 (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 A099503 (SageMath)
%o A099503 @CachedFunction
%o A099503 def a(n): # a = A099503
%o A099503     if (n<3): return 4^n
%o A099503     else: return 4*a(n-1) - a(n-3)
%o A099503 [a(n) for n in range(31)] # _G. C. Greubel_, Aug 03 2023
%Y A099503 Cf. A000071, A000302, A076264, A099504.
%K A099503 easy,nonn
%O A099503 0,2
%A A099503 _Paul Barry_, Oct 20 2004