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A154246 a(n) = ( (5 + sqrt(7))^n - (5 - sqrt(7))^n )/(2*sqrt(7)).

%I A154246 #30 Sep 08 2022 08:45:40
%S A154246 1,10,82,640,4924,37720,288568,2206720,16872976,129008800,986374432,
%T A154246 7541585920,57661119424,440862647680,3370726327168,25771735613440,
%U A154246 197044282245376,1506551581411840,11518718733701632,88069258871603200
%N A154246 a(n) = ( (5 + sqrt(7))^n - (5 - sqrt(7))^n )/(2*sqrt(7)).
%C A154246 Second binomial transform of A086901 without initial term 1.
%C A154246 Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(7) = 7.6457513110....
%H A154246 G. C. Greubel, <a href="/A154246/b154246.txt">Table of n, a(n) for n = 1..750</a>
%H A154246 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-18).
%F A154246 From _Philippe Deléham_, Jan 06 2009: (Start)
%F A154246 a(n) = 10*a(n-1) - 18*a(n-2) for n > 1, with a(0)=0, a(1)=1.
%F A154246 G.f.: x/(1 - 10*x + 18*x^2). (End)
%F A154246 E.g.f.: (1/sqrt(7))*exp(5*x)*sinh(sqrt(7)*x). - _G. C. Greubel_, Sep 07 2016
%t A154246 Table[Simplify[((5+Sqrt[7])^n -(5-Sqrt[7])^n)/(2*Sqrt[7])], {n,1,25}] (* or *) LinearRecurrence[{10, -18}, {1, 10}, 25] (* _G. C. Greubel_, Sep 07 2016 *)
%o A154246 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); 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 A154246 (Magma) I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-18*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Sep 08 2016
%o A154246 (Sage) [lucas_number1(n,10,18) for n in range(1, 25)] # _Zerinvary Lajos_, Apr 26 2009
%o A154246 (PARI) my(x='x+O('x^25)); Vec(x/(1-10*x+18*x^2)) \\ _G. C. Greubel_, May 31 2019
%Y A154246 Cf. A010465 (decimal expansion of square root of 7), A086901.
%K A154246 nonn,easy
%O A154246 1,2
%A A154246 Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%E A154246 Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009
%E A154246 Edited by _Klaus Brockhaus_, Oct 06 2009