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a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).
5

%I #26 Sep 08 2022 08:45:40

%S 1,8,54,352,2276,14688,94744,611072,3941136,25418368,163935584,

%T 1057300992,6819052096,43979406848,283644733824,1829363802112,

%U 11798463078656,76094066608128,490767902078464,3165202550546432

%N a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

%C Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(6) = 6.4494897427....

%C Binomial transform of A164550, second binomial transform of A164549, third binomial transform of A123011, fourth binomial transform of A164532.

%C Binomial transform is A164551, second binomial transform is A164552, third binomial transform is A164553.

%H G. C. Greubel, <a href="/A154235/b154235.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 (8,-10).

%F From _Philippe Deléham_, Jan 06 2009: (Start)

%F a(n) = 8*a(n-1) - 10*a(n-2) for n > 1, where a(0)=0, a(1)=1.

%F G.f.: x/(1 - 8*x + 10*x^2). (End)

%t LinearRecurrence[{8, -10}, {1, 8}, 30] (* or *) Table[Simplify[((4 + Sqrt[6])^n -(4-Sqrt[6])^n)/(2*Sqrt[6])], {n, 30}] (* _G. C. Greubel_, Sep 06 2016 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-6); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 07 2009

%o (Sage) [lucas_number1(n,8,10) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009

%o (PARI) a(n)=([0,1; -10,8]^(n-1)*[1;8])[1,1] \\ _Charles R Greathouse IV_, Sep 07 2016

%o (PARI) my(x='x+O('x^30)); Vec(x/(1-8*x+10*x^2)) \\ _G. C. Greubel_, May 21 2019

%o (GAP) a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-10*a[n-2]; od; a; # _G. C. Greubel_, May 21 2019

%Y Cf. A010464 (decimal expansion of square root of 6), A123011, A164532, A164549, A164550, A164551, A164552, A164553.

%K nonn,easy

%O 1,2

%A Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009

%E Edited by _Klaus Brockhaus_, Oct 04 2009