A153596 - OEIS

%I #25 Sep 08 2022 08:45:40

%S 1,10,78,560,3884,26520,179752,1214080,8186256,55152800,371430368,

%T 2500942080,16837952704,113358801280,763153053312,5137636904960,

%U 34587001876736,232842006858240,1567506027294208,10552536122060800,71040228620135424

%N a(n) = ((5 + sqrt(3))^n - (5 - sqrt(3))^n)/(2*sqrt(3)).

%C Third binomial transform of A054485. Fifth binomial transform of A162813 preceded by 1.

%C Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(3) = 6.73205080756887729....

%H G. C. Greubel, <a href="/A153596/b153596.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,-22).

%F G.f.: x/(1 - 10*x + 22*x^2). - _Klaus Brockhaus_, Dec 31 2008 [corrected Oct 11 2009]

%F a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0)=0, a(1)=1. - _Philippe Deléham_, Jan 01 2009

%F E.g.f.: sinh(sqrt(3)*x)*exp(5*x)/sqrt(3). - _Ilya Gutkovskiy_, Aug 23 2016

%t Table[Simplify[((5+Sqrt[3])^n -(5-Sqrt[3])^n)/(2*Sqrt[3])], {n,1,25}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2011, modified by _G. C. Greubel_, Jun 01 2019 *)

%t LinearRecurrence[{10,-22},{1,10},25] (* _G. C. Greubel_, Aug 22 2016 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Dec 31 2008

%o (Sage) [lucas_number1(n,10,22) for n in range(1, 25)] # _Zerinvary Lajos_, Apr 26 2009

%o (Magma) I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-22*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 23 2016

%o (PARI) my(x='x+O('x^25)); Vec(x/(1-10*x+22*x^2)) \\ _G. C. Greubel_, Jun 01 2019

%Y Cf. A002194 (decimal expansion of sqrt(3)), A054485, A162813.

%K nonn

%O 1,2

%A Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

%E Extended beyond a(7) by _Klaus Brockhaus_, Dec 31 2008

%E Edited by _Klaus Brockhaus_, Oct 11 2009