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A164545 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

%I A164545 #30 Sep 08 2022 08:45:47
%S A164545 1,8,36,176,848,4096,19776,95488,461056,2226176,10748928,51900416,
%T A164545 250597376,1209991168,5842354176,28209381376,136206942208,
%U A164545 657665294336,3175488946176,15332616962048,74032423632896,357460162379776
%N A164545 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
%C A164545 Binomial transform of A164544. Second binomial transform of A164640. Inverse binomial transform of A038761.
%H A164545 Harvey P. Dale, <a href="/A164545/b164545.txt">Table of n, a(n) for n = 0..1000</a> [extending from n(164) by Vincenzo Librandi]
%H A164545 Martin Burtscher, Igor Szczyrba and Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A164545 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).
%F A164545 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
%F A164545 a(n) = ((2+3*sqrt(2))*(2+2*sqrt(2))^n + (2-3*sqrt(2))*(2-2*sqrt(2))^n)/4.
%F A164545 G.f.: (1 + 4*x)/(1 - 4*x - 4*x^2).
%F A164545 a(n) = (2*i)^n*( ChebyshevU(n, -i) - 2*i*ChebyshevU(n-1, -i) ). - _G. C. Greubel_, Jul 17 2021
%t A164545 LinearRecurrence[{4,4},{1,8},30] (* _Harvey P. Dale_, Dec 25 2011 *)
%o A164545 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+3*r)*(2+2*r)^n+(2-3*r)*(2-2*r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 19 2009
%o A164545 (Sage) [(2*i)^n*(chebyshev_U(n, -i) - 2*i*chebyshev_U(n-1, -i)) for n in (0..30)] # _G. C. Greubel_, Jul 17 2021
%Y A164545 Cf. A038761, A164544, A164640.
%K A164545 nonn,easy
%O A164545 0,2
%A A164545 Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
%E A164545 Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 19 2009