A099462 - OEIS

%I #12 Sep 08 2022 08:45:15

%S 0,1,0,4,4,16,32,80,192,448,1088,2560,6144,14592,34816,82944,197632,

%T 471040,1122304,2674688,6373376,15187968,36192256,86245376,205520896,

%U 489750528,1167065088,2781085696,6627262464,15792603136,37633392640

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

%C Binomial transform is A099463.

%H G. C. Greubel, <a href="/A099462/b099462.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 (0,4,4).

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

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

%F a(n+1) = Sum_{k=0..floor(n/2)} C((n-k)/2, k)*(1+(-1)^(n-k))*2^(n-k). - _Paul Barry_, Sep 09 2005

%t LinearRecurrence[{0,4,4}, {0,1,0}, 40] (* _G. C. Greubel_, Nov 18 2021 *)

%o (Magma) [n le 3 select (1+(-1)^n)/2 else 4*(Self(n-2) +Self(n-3)): n in [1..41]]; // _G. C. Greubel_, Nov 18 2021

%o (Sage)

%o def a(n): return sum( 4^k*binomial(k, n-2*k-1) for k in (0..(n-1)//2) )

%o [a(n) for n in (0..40)] # _G. C. Greubel_, Nov 18 2021

%Y Cf. A099463.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Oct 16 2004