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%I #19 Sep 08 2022 08:45:47
%S 1,8,63,494,3869,30292,237147,1856506,14533561,113775008,890679543,
%T 6972620294,54584650709,427311962092,3345180558867,26187502233106,
%U 205006952830321,1604881990340408,12563701705288623,98354023217099294,769957303181086349
%N a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
%C Binomial transform of A164591. Fifth binomial transform of A096886.
%H G. C. Greubel, <a href="/A164592/b164592.txt">Table of n, a(n) for n = 0..1000</a> (* terms 0..144 from Vincenzo Librandi)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -17).
%F a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
%F a(n) = ((4 + sqrt(18))*(5 + sqrt(8))^n + (4 - sqrt(18))*(5 - sqrt(8))^n)/8.
%F G.f.: (1-2*x)/(1-10*x+17*x^2).
%F E.g.f.: (1/4)*exp(5*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 12 2017
%t CoefficientList[Series[(1 - 2*z)/(17*z^2 - 10*z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 12 2011 *)
%t LinearRecurrence[{10,-17},{1,8},30] (* _Harvey P. Dale_, Oct 14 2012 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((4+3*r)*(5+2*r)^n+(4-3*r)*(5-2*r)^n)/8: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 24 2009
%o (PARI) x='x+O('x^50); Vec((1-2*x)/(1-10*x+17*x^2)) \\ _G. C. Greubel_, Aug 12 2017
%Y Cf. A164591, A096886.
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
%E Extended by _Klaus Brockhaus_ and _R. J. Mathar_ Aug 24 2009
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