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%I #28 Sep 08 2022 08:45:40
%S 1,10,83,660,5189,40670,318487,2493480,19520521,152816050,1196311643,
%T 9365243580,73315137869,573942237830,4493065034527,35173632302160,
%U 275354217434641,2155590425209690,16874882555708003,132103788328515300
%N a(n) = ( (5 + 2*sqrt(2))^n - (5 - 2*sqrt(2))^n )/(4*sqrt(2)).
%C First differences are in A164588.
%C Lim_{n -> infinity} a(n)/a(n-1) = 5 + 2*sqrt(2) = 7.8284271247....
%H G. C. Greubel, <a href="/A154340/b154340.txt">Table of n, a(n) for n = 1..1000</a>
%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, with a(0)=0, a(1)=1. - _Philippe Deléham_, Jan 12 2009
%F G.f.: x/(1 - 10*x + 17*x^2). - _Klaus Brockhaus_, Jan 12 2009, corrected Oct 08 2009
%F E.g.f.: (1/sqrt(8))*exp(5*x)*sinh(2*sqrt(2)*x). - _G. C. Greubel_, Sep 11 2016
%p A154340:=n->((5+2*sqrt(2))^n-(5-2*sqrt(2))^n)/(4*sqrt(2)): seq(simplify(A154340(n)), n=1..30); # _Wesley Ivan Hurt_, Sep 12 2016
%t Drop[CoefficientList[Series[x/(1-10*x+17*x^2), {x,0,30}], x], 1] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2011, modified by _G. C. Greubel_, Jun 01 2019 *)
%t LinearRecurrence[{10, -17}, {1, 10}, 30] (* or *) Table[Simplify[((5 + 2*Sqrt[2])^n -(5-2*Sqrt[2])^n)/(4*Sqrt[2])], {n, 1, 30}] (* _G. C. Greubel_, Sep 11 2016 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)^n-(5-2*r)^n)/(4*r): n in [1..30] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 12 2009
%o (Sage) [lucas_number1(n,10,17) for n in range(1, 30)] # _Zerinvary Lajos_, Apr 26 2009
%o (Magma) I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-17*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 12 2016
%o (PARI) a(n)=([0,1; -17,10]^(n-1)*[1;10])[1,1] \\ _Charles R Greathouse IV_, Sep 12 2016
%Y Cf. A002193 (decimal expansion of sqrt(2)), A164588.
%K nonn,easy
%O 1,2
%A Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 12 2009
%E Edited by _Klaus Brockhaus_, Oct 08 2009
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