%I #56 Feb 08 2024 07:15:07
%S 0,1,2,-6,-32,-4,312,664,-1792,-10224,-2528,97184,219648,-532544,
%T -3261568,-1197696,30220288,72417536,-157367808,-1038910976,
%U -504143872,9380822016,23803082752,-46202054656,-330434936832,-198849327104,2906650714112,7801794699264
%N a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.
%C For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - _G. C. Greubel_, Jun 10 2022
%H Vincenzo Librandi, <a href="/A190958/b190958.txt">Table of n, a(n) for n = 0..190</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-10).
%F G.f.: x / ( 1 - 2*x + 10*x^2 ). - _R. J. Mathar_, Jun 01 2011
%F E.g.f.: (1/3)*exp(x)*sin(3*x). - _Franck Maminirina Ramaharo_, Nov 13 2018
%F a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - _G. C. Greubel_, Jun 10 2022
%F a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - _Gerry Martens_, Oct 15 2022
%t LinearRecurrence[{2,-10}, {0,1}, 50]
%o (Magma) I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 17 2011
%o (PARI) a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ _Charles R Greathouse IV_, Apr 08 2016
%o (SageMath) [lucas_number1(n,2,10) for n in (0..50)] # _G. C. Greubel_, Jun 10 2022
%Y Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
%Y c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
%Y 1..A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
%Y 2..A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
%Y 3..A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
%Y 4..A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
%Y 5..A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
%Y 6..A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
%Y 7..A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
%Y 8..A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
%Y 9..A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
%Y 10.A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086
%K sign,easy
%O 0,3
%A _Vladimir Joseph Stephan Orlovsky_, May 24 2011