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a(n) = ((5+sqrt(2))*(1+sqrt(2))^n + (5-sqrt(2))*(1-sqrt(2))^n)/2.
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%I #20 Mar 15 2024 02:23:20

%S 5,7,19,45,109,263,635,1533,3701,8935,21571,52077,125725,303527,

%T 732779,1769085,4270949,10310983,24892915,60096813,145086541,

%U 350269895,845626331,2041522557,4928671445,11898865447,28726402339,69351670125

%N a(n) = ((5+sqrt(2))*(1+sqrt(2))^n + (5-sqrt(2))*(1-sqrt(2))^n)/2.

%C Binomial transform of A162396.

%H Vincenzo Librandi, <a href="/A162268/b162268.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1)

%F a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 5, a(1) = 7.

%F G.f.: (5-3*x)/(1-2*x-x^2).

%F a(n) = 5*A000129(n+1) - 3*A000129(n). - _R. J. Mathar_, Mar 06 2013

%F a(n) = 4*A001333(n) + A001333(n+1). - _G. C. Greubel_, Aug 17 2018

%t LinearRecurrence[{2,1}, {5,7}, 30] (* _Vincenzo Librandi_, Feb 03 2018 *)

%t Table[(4*LucasL[n, 2] + LucasL[n + 1, 2])/2, {n, 0, 30}] (* _G. C. Greubel_, Aug 17 2018 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+r)*(1+r)^n+(5-r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 02 2009

%o (PARI) x='x+O('x^30); Vec((5-3*x)/(1-2*x-x^2)) \\ _G. C. Greubel_, Aug 17 2018

%Y Cf. A162396.

%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 02 2009