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%I #28 Mar 04 2024 09:01:56
%S 2,76,5778,439204,33385282,2537720636,192900153618,14662949395604,
%T 1114577054219522,84722519070079276,6440026026380244498,
%U 489526700523968661124,37210469265847998489922,2828485190904971853895196,215002084978043708894524818,16342986943522226847837781364,1242282009792667284144565908482
%N a(n) = Lucas(9*n).
%C a(n+1)/a(n) converges to (76 + sqrt(5780))/2 = 76.01315561749...
%C a(0)/a(1) = 2/76, a(1)/a(2) = 76/5778, a(2)/a(3) = 5778/439204, a(3)/a(4) = 439204/33385282, etc.
%C Lim_{n->infinity} a(n)/a(n+1) = 0.01315561749... = 2/(76 + sqrt(5780)) = (sqrt(5780) - 76)/2.
%H Indranil Ghosh, <a href="/A087287/b087287.txt">Table of n, a(n) for n = 0..530</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 (76,1).
%F a(n) = 76a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 76.
%F a(n) = ((76 + sqrt(5780))/2)^n + ((76 - sqrt(5780))/2)^n.
%F a(n)^2 = a(2n) - 2 for n = 1, 3, 5, ...;
%F a(n)^2 = a(2n) + 2 for n = 2, 4, 6, ....
%F G.f.: (2-76*x)/(1-76*x-x^2). - _Philippe Deléham_, Nov 02 2008
%e a(4) = 33385282 = 76*a(3) + a(2) = 76*439204 + 5778 = ((76 + sqrt(5780))/2)^4 + ((76 - sqrt(5780))/2)^4 = 33385281.999999970046... + 0.000000029953... = 33385282.
%t LucasL[9*Range[0, 20]] (* _Paolo Xausa_, Mar 04 2024 *)
%o (Magma) [ Lucas(9*n) : n in [0..100]]; // _Vincenzo Librandi_, Apr 14 2011
%o (PARI) a(n)=fibonacci(9*n-1)+fibonacci(9*n+1) \\ _Charles R Greathouse IV_, Feb 06 2017
%Y Cf. A000032.
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003
%E More terms from _Vincenzo Librandi_, Apr 14 2011
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