A162275 - OEIS

%I #18 Sep 08 2022 08:45:46

%S 2,13,86,574,3848,25852,173864,1169896,7873952,53001808,356791136,

%T 2401871584,16169310848,108851933632,732794497664,4933202436736,

%U 33210545418752,223575000579328,1505118006580736,10132530053062144,68212704385845248,459211382691085312

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

%C Binomial transform of A162274.

%H Vincenzo Librandi, <a href="/A162275/b162275.txt">Table of n, a(n) for n = 0..149</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -22).

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

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

%F G.f.: (2-7*x)/(1-10*x+22*x^2).

%p 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 CoefficientList[Series[(2 - 7 z)/(22 z^2 - 10 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 12 2011 *)

%t LinearRecurrence[{10,-22},{2,13},30] (* _Harvey P. Dale_, Jun 14 2017 *)

%o (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 Cf. 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 _Klaus Brockhaus_, Jul 05 2009