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A164605 a(n) = ((1+4*sqrt(2))*(4+2*sqrt(2))^n + (1-4*sqrt(2))*(4-2*sqrt(2))^n)/2.

%I A164605 #18 Sep 08 2022 08:45:47
%S A164605 1,20,152,1056,7232,49408,337408,2304000,15732736,107429888,733577216,
%T A164605 5009178624,34204811264,233565061120,1594881998848,10890535501824,
%U A164605 74365228023808,507797540175872,3467458497216512,23677287656325120
%N A164605 a(n) = ((1+4*sqrt(2))*(4+2*sqrt(2))^n + (1-4*sqrt(2))*(4-2*sqrt(2))^n)/2.
%C A164605 Binomial transform of A164604. Fourth binomial transform of A164702. Inverse binomial transform of A164606.
%H A164605 G. C. Greubel, <a href="/A164605/b164605.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..149 from Vincenzo Librandi)
%H A164605 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8).
%F A164605 a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 20.
%F A164605 G.f.: (1+12*x)/(1-8*x+8*x^2).
%F A164605 E.g.f.: exp(4*x)*(cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 10 2017
%t A164605 LinearRecurrence[{8,-8},{1,20},30] (* _Harvey P. Dale_, Mar 24 2015 *)
%o A164605 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+2*r)^n+(1-4*r)*(4-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 23 2009
%o A164605 (PARI) Vec((1+12*x)/(1-8*x+8*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 12 2011
%Y A164605 Cf. A164604, A164702, A164606.
%K A164605 nonn,easy
%O A164605 0,2
%A A164605 Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
%E A164605 Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 23 2009