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%I #51 Dec 03 2023 12:34:45
%S 1,2,12,56,272,1312,6336,30592,147712,713216,3443712,16627712,
%T 80285696,387653632,1871757312,9037643776,43637604352,210700992512,
%U 1017354387456,4912221519872,23718303629312,114522100596736,552961616904192,2669934870003712
%N a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
%C Original name was: Generalized Fibonacci sequence.
%C Binomial transform of A084058.
%H Vincenzo Librandi, <a href="/A084128/b084128.txt">Table of n, a(n) for n = 0..1000</a>
%H Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="http://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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).
%F a(n) = 2^n * A001333(n).
%F G.f.: (1-2*x)/(1-4*x-4*x^2).
%F a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
%F a(n) = (2 + 2*sqrt(2))^n/2 + (2 - 2*sqrt(2))^n/2.
%F E.g.f.: exp(2*x)*cosh(2*x*sqrt(2)).
%F From _Johannes W. Meijer_, Aug 01 2010: (Start)
%F Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2).
%F Lim_{n->infinity} A084128(n)/A057087(n-1) = sqrt(2). (End)
%F a(n) = Sum_{k=0..n} A201730(n,k)*7^k. - _Philippe Deléham_, Dec 06 2011
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k-2)/(x*(4*k+2) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013
%F a(n) = 2^(n-1)*A002203(n). - _Vladimir Reshetnikov_, Oct 07 2016
%p a:=proc(n) option remember; if n=0 then 1 elif n=1 then 2 else
%p 4*a(n-1)+4*a(n-2); fi; end: seq(a(n), n=0..40); # _Wesley Ivan Hurt_, Jan 31 2017
%p a := n -> (2*I)^n*ChebyshevT(n, -I):
%p seq(simplify(a(n)), n = 0..23); # _Peter Luschny_, Dec 03 2023
%t CoefficientList[Series[(2 z - 1)/(4 z^2 + 4 z - 1), {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 01 2011 *)
%t Table[2^(n-1) LucasL[n, 2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 07 2016 *)
%t LinearRecurrence[{4,4},{1,2},30] (* _Harvey P. Dale_, Mar 01 2018 *)
%o (PARI) a(n)=if(n<0,0,polsym(4+4*x-x^2,n)[n+1]/2)
%o (Sage) [lucas_number2(n,4,-4)/2 for n in range(0, 23)] # _Zerinvary Lajos_, May 14 2009
%o (Magma) [2^(n-1)*Evaluate(DicksonFirst(n,-1), 2): n in [0..40]]; // _G. C. Greubel_, Oct 13 2022
%Y Cf. A001333 A001541, A002203, A057087, A084058, A084128, A201730.
%Y Appears in A086346, A086347 and A086348. - _Johannes W. Meijer_, Aug 01 2010
%K nonn,easy
%O 0,2
%A _Paul Barry_, May 16 2003
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