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%I #31 Sep 08 2022 08:45:12
%S 2,199,39603,7881196,1568397607,312119004989,62113250390418,
%T 12360848946698171,2459871053643326447,489526700523968661124,
%U 97418273275323406890123,19386725908489881939795601,3858055874062761829426214722
%N a(n) = Lucas(11*n).
%C Lim_{n-> infinity} a(n+1)/a(n) = 199.00502499874... = (199 + sqrt(39605))/2.
%C Lim_{n->infinity} a(n)/a(n+1) = 0.00502499874... = 2/(199 + sqrt(39605)) = (sqrt(39605) - 199)/2.
%H Nathaniel Johnston, <a href="/A089772/b089772.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 (199,1).
%F a(n) = 199*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 199.
%F a(n) = ((199 + sqrt(39605))/2)^n + ((199 - sqrt(39605))/2)^n.
%F a(n)^2 = a(2n) - 2 if n = 1, 3, 5, ...;
%F a(n)^2 = a(2n) + 2 if n = 2, 4, 6, ....
%F G.f.: (2 - 199*x)/(1 - 199*x - x^2). - _Philippe Deléham_, Nov 02 2008
%F a(n) = Lucas(n, 199) = 2*(-i)^n * ChebyshevT(n, 199*i/2). - _G. C. Greubel_, Dec 31 2019
%F E.g.f.: 2*exp(199*x/2)*cosh(sqrt(39605)*x/2). - _Stefano Spezia_, Jan 01 2020
%e a(4) = 1568397607 = 199*a(3) + a(2) = 199*7881196 + 39603 = ((199 + sqrt(39605) )/2)^4 + ((199 - sqrt(39605))/2)^4 = 1568397606.9999999993624065... + 0.0000000006375934...
%p seq(simplify(2*(-I)^n*ChebyshevT(n, 199*I/2)), n = 0..20); # _G. C. Greubel_, Dec 31 2019
%t LucasL[11*Range[0,20]] (* or *) LinearRecurrence[{199,1},{2,199},20] (* _Harvey P. Dale_, Dec 23 2015 *)
%t LucasL[Range[20]-1,199] (* _G. C. Greubel_, Dec 31 2019 *)
%o (Magma) [Lucas(11*n): n in [0..20]]; // _Vincenzo Librandi_, Apr 15 2011
%o (PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 199*I/2) ) \\ _G. C. Greubel_, Dec 31 2019
%o (Sage) [lucas_number2(11*n,1,-1) for n in (0..20)] # _G. C. Greubel_, Dec 30 2019
%o (GAP) List([0..20], n-> Lucas(1,-1,11*n)[2] ); # _G. C. Greubel_, Dec 30 2019
%Y Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25), A087281 (m=29), A087287 (m=76), this sequence (m=199).
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 09 2004
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