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(4^n+2^n+0^n+(-2)^n)/4
2

%I #13 Sep 08 2022 08:45:10

%S 1,1,6,16,72,256,1056,4096,16512,65536,262656,1048576,4196352,

%T 16777216,67117056,268435456,1073774592,4294967296,17180000256,

%U 68719476736,274878431232,1099511627776,4398048608256,17592186044416,70368752566272

%N (4^n+2^n+0^n+(-2)^n)/4

%C Binomial transform of A083884.

%H Vincenzo Librandi, <a href="/A083885/b083885.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16).

%F a(n) = (4^n+2^n+0^n+(-2)^n)/4.

%F G.f.: (4*x^3-2*x^2-3*x+1)/((2*x+1)*(2*x-1)*(4*x-1)).

%F E.g.f.: exp(4*x)+exp(2*x)+exp(0)+exp(-2*x).

%F A007814(a(n)) = A022998(n-1). - _Ralf Stephan_, Feb 14 2004

%F a(0)=1, a(1)=1, a(2)=6, a(3)=16, a(n)=4*a(n-1)+4*a(n-2)-16*a(n-3) [From _Harvey P. Dale_, Dec 12 2011]

%t Join[{1},Table[(4^n+2^n+(-2)^n)/4,{n,30}]] (* or *) Join[{1}, LinearRecurrence[ {4,4,-16},{1,6,16},30]] (* _Harvey P. Dale_, Dec 12 2011 *)

%o (Magma) [(4^n+2^n+0^n+(-2)^n)/4: n in [0..20]]; // _Vincenzo Librandi_, Jun 16 2011

%K easy,nonn

%O 0,3

%A _Paul Barry_, May 09 2003