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Lucas numbers L(12n).
9

%I #39 Sep 08 2022 08:45:12

%S 2,322,103682,33385282,10749957122,3461452808002,1114577054219522,

%T 358890350005878082,115561578124838522882,37210469265847998489922,

%U 11981655542024930675232002,3858055874062761829426214722,1242282009792667284144565908482,400010949097364802732720796316482

%N Lucas numbers L(12n).

%C a(n+1)/a(n) converges to (322 + sqrt(103680))/2 = 321.996894379... a(0)/a(1) = 2/322; a(1)/a(2) = 322/103682; a(2)/a(3) = 103682/33385282; a(3)/a(4) = 33385282/10749957122; etc. Lim_{n -> inf} a(n)/a(n+1) = 0.00310562... = 2/(322 + sqrt(103680)) = (322 - sqrt(103680))/2.

%H Nathaniel Johnston, <a href="/A089775/b089775.txt">Table of n, a(n) for n = 0..100</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (322, -1).

%F a(n) = 322*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 322

%F a(n) = ((322 + sqrt(103680))/2)^n + ((322 - sqrt(103680))/2)^n.

%F (a(n))^2 = a(2n) + 2.

%F G.f.: (2-322*x)/(1-322*x+x^2). - _Philippe Deléham_, Nov 02 2008

%e a(4) = 10749957122 = 322*a(3) - a(2) = 322*33385282 - 103682 = ((322 + sqrt(103680))/2)^4 + ((322 - sqrt(103680))/2)^4.

%t Table[LucasL[12n], {n, 0, 13}] (* _Indranil Ghosh_, Mar 15 2017 *)

%o (Magma) [ Lucas(12*n) : n in [0..70]]; // _Vincenzo Librandi_, Apr 15 2011

%o (PARI) Vec((2 - 322*x)/(1 - 322*x + x^2) + O(x^14)) \\ _Indranil Ghosh_, Mar 15 2017

%Y Cf. A000032, A060964.

%Y a(n) = A000032(12n).

%Y Row 9 * 2 of array A188644

%K easy,nonn

%O 0,1

%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 09 2004

%E a(11) - a(13) from _Vincenzo Librandi_, Apr 15 2011