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A162273 a(n) = ((2+sqrt(3))*(3+sqrt(3))^n + (2-sqrt(3))*(3-sqrt(3))^n)/2. 1

%I #17 Sep 08 2022 08:45:46

%S 2,9,42,198,936,4428,20952,99144,469152,2220048,10505376,49711968,

%T 235239552,1113165504,5267555712,24926341248,117952713216,

%U 558158231808,2641233111552,12498449278464,59143297001472,279869086338048

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

%C Binomial transform of A001075 without initial term 1, inverse binomial transform of A162274.

%C The INVERTi transform yields A007051 without A007051(0). - _R. J. Mathar_, Jul 07 2009

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).

%F a(n) = 6*a(n-1) - 6*a(n-2) for n > 1; a(0) = 2, a(1) = 9.

%F G.f.: (2-3*x)/(1-6*x+6*x^2).

%F a(n) = 2*A030192-3*A030192(n-1). - _R. J. Mathar_, Feb 04 2021

%p seq(simplify(((2+sqrt(3))*(3+sqrt(3))^n+(2-sqrt(3))*(3-sqrt(3))^n)*1/2), n = 0 .. 22); # _Emeric Deutsch_, Jul 11 2009

%t LinearRecurrence[{6,-6},{2,9},30] (* _Harvey P. Dale_, Dec 17 2019 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((2+r)*(3+r)^n+(2-r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 05 2009

%Y Cf. A001075, A162274.

%K nonn,easy

%O 0,1

%A Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

%E Edited and extended beyond a(5) by _R. J. Mathar_ and _Klaus Brockhaus_, Jul 05 2009

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Last modified March 29 09:28 EDT 2024. Contains 371268 sequences. (Running on oeis4.)