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%I #32 Mar 08 2021 11:57:02
%S 1,3,14,52,216,848,3424,13632,54656,218368,873984,3494912,13981696,
%T 55922688,223698944,894779392,3579150336,14316535808,57266274304,
%U 229064835072,916259864576,3665038409728,14660155736064,58640618749952
%N a(n) = (5*4^n + (-2)^n)/6.
%C Binomial transform of A083423.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,8).
%F a(n) = 2*a(n-1) + 8*a(n-2). - _N. J. A. Sloane_, Jul 16 2014
%F G.f.: (1+x)/(1-2*x-8*x^2). [Corrected by _N. J. A. Sloane_, Jul 16 2014]
%F E.g.f.: (5*exp(4*x) + exp(-2*x))/6.
%F From _N. J. A. Sloane_, Jul 18 2014: (Start)
%F 2^(n-1)|a(n) for n >= 1;
%F 3|a(3n+1). (End)
%F From _Klaus Purath_, Oct 15 2020: (Start)
%F a(n) = A048573(n)*2^(n-1).
%F a(n) = A048573(n)*(A048573(n+1) - A048573(n-1))/5. (End)
%e Factorizations of initial terms: 1, (3), (2)*(7), (2)^2*(13), (2)^3*(3)^3, (2)^4*(53), (2)^5*(107), (2)^6*(3)*(71), (2)^7*(7)*(61), (2)^8*(853), (2)^9*(3)*(569), (2)^10*(3413), (2)^11*(6827), (2)^12*(3)^2*(37)*(41), (2)^13*(7)*(47)*(83), (2)^14*(13)*(4201), (2)^15*(3)*(23)*(1583), (2)^16*(218453), ...
%p A083424:=n->(5*4^n+(-2)^n)/6; [seq(A083424(n),n=0..50)]; # _N. J. A. Sloane_, Jul 18 2014
%t LinearRecurrence[{2,8},{1,3},30] (* _Harvey P. Dale_, Apr 21 2019 *)
%o (PARI) a(n)=(5*4^n+(-2)^n)/6 \\ _Charles R Greathouse IV_, Sep 24 2015
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 30 2003
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