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%I #23 Sep 08 2022 08:45:40
%S 1,10,81,620,4661,34830,259741,1935640,14421321,107436050,800355401,
%T 5962269060,44415937981,330876267670,2464859855061,18361949464880,
%U 136787157402641,1018994534193690,7590989351286721,56548997363187100
%N a(n) = ( (5 + sqrt(6))^n - (5 - sqrt(6))^n )/(2*sqrt(6)).
%C First differences are in A164551.
%C Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(6) = 7.4494897427....
%H G. C. Greubel, <a href="/A154236/b154236.txt">Table of n, a(n) for n = 1..999</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -19).
%F From _Philippe Deléham_, Jan 06 2009: (Start)
%F a(n) = 10*a(n-1) - 19*a(n-2) for n > 1, with a(0)=0, a(1)=1.
%F G.f.: x/(1 - 10*x + 19*x^2). (End)
%t LinearRecurrence[{10, -19}, {1, 10}, 25] (* or *) Table[Simplify[((5 + Sqrt[6])^n -(5-Sqrt[6])^n)/(2*Sqrt[6])], {n, 1, 25}] (* _G. C. Greubel_, Sep 06 2016 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-6); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 07 2009
%o (Sage) [lucas_number1(n,10,19) for n in range(1, 25)] # _Zerinvary Lajos_, Apr 26 2009
%o (PARI) a(n)=([0,1; -19,10]^(n-1)*[1;10])[1,1] \\ _Charles R Greathouse IV_, Sep 07 2016
%Y Cf. A010464 (decimal expansion of square root of 6), A164551.
%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