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A055819 Row sums of array T in A055818; twice the odd-indexed Fibonacci numbers after initial term.

%I A055819 #46 Sep 08 2022 08:45:01
%S A055819 1,2,4,10,26,68,178,466,1220,3194,8362,21892,57314,150050,392836,
%T A055819 1028458,2692538,7049156,18454930,48315634,126491972,331160282,
%U A055819 866988874,2269806340,5942430146,15557484098,40730022148,106632582346
%N A055819 Row sums of array T in A055818; twice the odd-indexed Fibonacci numbers after initial term.
%C A055819 Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy - 4. - _Michel Lagneau_, Feb 01 2014
%C A055819 Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 256 = 0. - _Colin Barker_, Feb 16 2014
%H A055819 Vincenzo Librandi, <a href="/A055819/b055819.txt">Table of n, a(n) for n = 0..1000</a>
%H A055819 Youngwoo Kwon, <a href="https://arxiv.org/abs/1804.08119">Binomial transforms of the modified k-Fibonacci-like sequence</a>, arXiv:1804.08119 [math.NT], 2018.
%H A055819 D. Yaqubi, M. Farrokhi D.G., H. Gahsemian Zoeram, <a href="https://arxiv.org/abs/1612.08697">Lattice paths inside a table. I</a>, arXiv:1612.08697 [math.CO], 2016-2017.
%H A055819 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F A055819 From _Colin Barker_, Feb 01 2014: (Start)
%F A055819 a(n) = 3*a(n-1) - a(n-2) for n > 0.
%F A055819 G.f.: (1 -x -x^2)/(1-3*x+x^2). (End)
%F A055819 a(n) = 2*A001519(n) for n > 0. - _Colin Barker_, Feb 04 2014
%F A055819 From _G. C. Greubel_, Jan 22 2020: (Start)
%F A055819 a(n) = 2*(ChebyshevU(n, 3/2) - 2*ChebyshevU(n-1, 3/2)), with a(0)=1.
%F A055819 E.g.f.: -1 + 2*exp(3*x/2)*( cosh(sqrt(5)*x/2) - sinh(sqrt(5)*x/2)/sqrt(5) ). (End)
%p A055819 seq(`if`(n=0, 1, simplify(2*(ChebyshevU(n, 3/2)-2*ChebyshevU(n-1, 3/2)))), n = 0..30); # _G. C. Greubel_, Jan 22 2020
%t A055819 CoefficientList[Series[(1-x-x^2)/(1-3*x+x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 05 2014 *)
%t A055819 Join[{1},LinearRecurrence[{3,-1},{2,4},30]] (* _Harvey P. Dale_, Oct 01 2014 *)
%t A055819 Table[If[n==0, 1, 2*(ChebyshevU[n, 3/2] -2*ChebyshevU[n-1, 3/2])], {n,0,30}] (* _G. C. Greubel_, Jan 22 2020 *)
%o A055819 (PARI) Vec((1-x-x^2)/(1-3*x+x^2) + O(x^40)) \\ _Colin Barker_, Feb 01 2014
%o A055819 (Magma) I:=[2,4]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 22 2020
%o A055819 (Sage) [1]+[2*(chebyshev_U(n,3/2) -2*chebyshev_U(n-1,3/2)) for n in (1..30)] # _G. C. Greubel_, Jan 22 2020
%o A055819 (GAP) a:=[2,4];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Concatenation([1], a); # _G. C. Greubel_, Jan 22 2020
%Y A055819 Essentially the same as A052995.
%K A055819 nonn,easy
%O A055819 0,2
%A A055819 _Clark Kimberling_, May 28 2000