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A090017 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1.

%I A090017 #56 Sep 08 2022 08:45:12
%S A090017 0,1,4,18,80,356,1584,7048,31360,139536,620864,2762528,12291840,
%T A090017 54692416,243353344,1082798208,4817899520,21437194496,95384577024,
%U A090017 424412697088,1888419942400,8402505163776,37386860539904,166352452487168
%N A090017 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1.
%C A090017 Starting with "1" = INVERT transform of A007482: (1, 3, 11, 39, 139, ...). - _Gary W. Adamson_, Aug 06 2010
%C A090017 This is the Lucas sequence U(4,-2). - _Bruno Berselli_, Jan 09 2013
%C A090017 Lower left term in matrix powers of [(1,5); (1,3)]. Convolved with (1, 2, 0, 0, 0, ...) the result is A164549: (1, 6, 26, 116, ...). - _Gary W. Adamson_, Aug 10 2016
%C A090017 For n>0, a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} in which 0 and 1 avoid runs of odd lengths. - _Milan Janjic_, Jan 08 2017
%H A090017 Vincenzo Librandi, <a href="/A090017/b090017.txt">Table of n, a(n) for n = 0..1000</a>
%H A090017 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.
%H A090017 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A090017 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,2).
%F A090017 G.f.: x/(1 - 4*x - 2*x^2).
%F A090017 a(n) = (-i*sqrt(2))^(n-1) U(n-1, i*sqrt(2)) where U is the Chebyshev polynomial of the second kind and i^2 = -1.
%F A090017 a(n) = ((2+sqrt(6))^n - (2-sqrt(6))^n)/(2 sqrt(6)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009, Jan 07 2009
%F A090017 a(n+1) = Sum_{k=0..n} A099089(n,k)*2^k. - _Philippe Deléham_, Nov 21 2011
%F A090017 From _Ilya Gutkovskiy_, Aug 22 2016: (Start)
%F A090017 E.g.f.: sinh(sqrt(6)*x)*exp(2*x)/sqrt(6).
%F A090017 Number of zeros in substitution system {0 -> 11, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 1101111011111101111011 -> ...). (End)
%t A090017 a[n_Integer] := (-I Sqrt[2])^(n - 1) ChebyshevU[ n - 1, I Sqrt[2] ]
%t A090017 a[n_]:=(MatrixPower[{{1,5},{1,3}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t A090017 t={0,1};Do[AppendTo[t,4*t[[-1]]+2*t[[-2]]],{n,2,23}];t (* or *) LinearRecurrence[{4,2},{0,1},24] (* _Indranil Ghosh_, Feb 21 2017 *)
%o A090017 (Sage) [lucas_number1(n, 4, -2) for n in range(0, 23)] # _Zerinvary Lajos_, Apr 23 2009
%o A090017 (Magma) I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 12 2011
%o A090017 (PARI) Vec(x/(1-4*x-2*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 12 2011
%Y A090017 Cf. A007070, A084059, A007482.
%Y A090017 Cf. A164549.
%K A090017 nonn,easy
%O A090017 0,3
%A A090017 _Paul Barry_, Nov 19 2003
%E A090017 Edited by _Stuart Clary_, Oct 25 2009