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A162275 a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 2, a(1) = 13.

%I A162275 #18 Sep 08 2022 08:45:46
%S A162275 2,13,86,574,3848,25852,173864,1169896,7873952,53001808,356791136,
%T A162275 2401871584,16169310848,108851933632,732794497664,4933202436736,
%U A162275 33210545418752,223575000579328,1505118006580736,10132530053062144,68212704385845248,459211382691085312
%N A162275 a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 2, a(1) = 13.
%C A162275 Binomial transform of A162274.
%H A162275 Vincenzo Librandi, <a href="/A162275/b162275.txt">Table of n, a(n) for n = 0..149</a>
%H A162275 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -22).
%F A162275 a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 2, a(1) = 13.
%F A162275 a(n) = ((2+sqrt(3))*(5+sqrt(3))^n + (2-sqrt(3))*(5-sqrt(3))^n)/2.
%F A162275 G.f.: (2-7*x)/(1-10*x+22*x^2).
%p A162275 a := proc (n) options operator, arrow; expand((1/2)*(2+sqrt(3))*(5+sqrt(3))^n+(1/2)*(2-sqrt(3))*(5-sqrt(3))^n) end proc: seq(a(n), n = 0 .. 20); # _Emeric Deutsch_, Jul 09 2009
%t A162275 CoefficientList[Series[(2 - 7 z)/(22 z^2 - 10 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 12 2011 *)
%t A162275 LinearRecurrence[{10,-22},{2,13},30] (* _Harvey P. Dale_, Jun 14 2017 *)
%o A162275 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((2+r)*(5+r)^n+(2-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 05 2009
%Y A162275 Cf. A162274.
%K A162275 nonn,easy
%O A162275 0,1
%A A162275 Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009
%E A162275 Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 05 2009