|
%I #18 Sep 08 2022 08:45:12
%S 2,19,363,6916,131767,2510489,47831058,911300591,17362542287,
%T 330799604044,6302555019123,120079344967381,2287810109399362,
%U 43588471423555259,830468767156949283,15822495047405591636
%N a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.
%C Lim_{n-> infinity} a(n)/a(n+1) = 0.052486... = 2/(19+sqrt(365)) = (sqrt(365)-19)/2.
%C Lim_{n-> infinity} a(n+1)/a(n) = 19.052486... = (19+sqrt(365))/2 = 2/(sqrt(365)-19).
%H G. C. Greubel, <a href="/A090308/b090308.txt">Table of n, a(n) for n = 0..500</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 (19,1).
%F a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.
%F a(n) = ((19+sqrt(365))/2)^n + ((19-sqrt(365))/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.: (2-19*x)/(1-19*x-x^2). - _Philippe Deléham_, Nov 02 2008
%F a(n) = Lucas(n, 19) = 2*(-i)^n * ChebyshevT(n, 19*i/2). - _G. C. Greubel_, Dec 30 2019
%e a(4) = 19*a(3) + a(2) = 19*6916 + 363 = ((19+sqrt(365))/2)^4 + ((19-sqrt(365))/2)^4 = 131766.9999924108 + 0.0000075891 = 131767.
%p seq(simplify(2*(-I)^n*ChebyshevT(n, 19*I/2)), n = 0..20); # _G. C. Greubel_, Dec 30 2019
%t LucasL[Range[20]-1,20] (* _G. C. Greubel_, Dec 30 2019 *)
%o (PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 19*I/2) ) \\ _G. C. Greubel_, Dec 30 2019
%o (Magma) m:=19; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 30 2019
%o (Sage) [2*(-I)^n*chebyshev_T(n, 19*I/2) for n in (0..20)] # _G. C. Greubel_, Dec 30 2019
%o (GAP) m:=19;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # _G. C. Greubel_, Dec 30 2019
%Y Cf. A049270.
%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), this sequence (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004
%E More terms from _Ray Chandler_, Feb 14 2004
|
