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A087287 a(n) = Lucas(9*n).

%I A087287 #33 Jan 18 2025 21:57:10
%S A087287 2,76,5778,439204,33385282,2537720636,192900153618,14662949395604,
%T A087287 1114577054219522,84722519070079276,6440026026380244498,
%U A087287 489526700523968661124,37210469265847998489922,2828485190904971853895196,215002084978043708894524818,16342986943522226847837781364,1242282009792667284144565908482
%N A087287 a(n) = Lucas(9*n).
%C A087287 a(n+1)/a(n) converges to (76 + sqrt(5780))/2 = 76.01315561749...
%C A087287 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 A087287 Lim_{n->oo} a(n)/a(n+1) = 0.01315561749... = 2/(76 + sqrt(5780)) = (sqrt(5780) - 76)/2.
%H A087287 Indranil Ghosh, <a href="/A087287/b087287.txt">Table of n, a(n) for n = 0..530</a>
%H A087287 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A087287 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A087287 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (76,1).
%F A087287 a(n) = 76a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 76.
%F A087287 a(n) = ((76 + sqrt(5780))/2)^n + ((76 - sqrt(5780))/2)^n.
%F A087287 a(n)^2 = a(2n) - 2 for n = 1, 3, 5, ...;
%F A087287 a(n)^2 = a(2n) + 2 for n = 2, 4, 6, ....
%F A087287 G.f.: (2-76*x)/(1-76*x-x^2). - _Philippe Deléham_, Nov 02 2008
%F A087287 E.g.f.: 2*exp(38*x)*cosh(17*sqrt(5)*x). - _Stefano Spezia_, Jan 18 2025
%e A087287 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 A087287 LucasL[9*Range[0, 20]] (* _Paolo Xausa_, Mar 04 2024 *)
%o A087287 (Magma) [ Lucas(9*n) : n in [0..100]]; // _Vincenzo Librandi_, Apr 14 2011
%o A087287 (PARI) a(n)=fibonacci(9*n-1)+fibonacci(9*n+1) \\ _Charles R Greathouse IV_, Feb 06 2017
%Y A087287 Cf. A000032.
%K A087287 easy,nonn
%O A087287 0,1
%A A087287 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003
%E A087287 More terms from _Vincenzo Librandi_, Apr 14 2011