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%I #23 Sep 08 2022 08:45:12
%S 2,23,531,12236,281959,6497293,149719698,3450050347,79500877679,
%T 1831970236964,42214816327851,972772745777537,22415987969211202,
%U 516540496037635183,11902847396834820411,274282030623238504636,6320389551731320427039,145643241720443608326533,3356114949121934311937298
%N a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
%C Lim_{n -> infinity} a(n)/a(n+1) = 0.04339638... = 2/(23+sqrt(533)) = (sqrt(533)-23)/2.
%C Lim_{n -> infinity} a(n+1)/a(n) = 23.04339638... = (23+sqrt(533))/2 = 2/(sqrt(533) - 23).
%H G. C. Greubel, <a href="/A090314/b090314.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 (23,1).
%F a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
%F a(n) = ((23 + sqrt(533))/2)^n + ((23 - sqrt(533))/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.: (2-23*x)/(1-23*x-x^2). - _Philippe Deléham_, Nov 02 2008
%F a(n) = Lucas(n, 23) = 2*(-i)^n * ChebyshevT(n, 23*i/2). - _G. C. Greubel_, Dec 29 2019
%e a(4) = 281959 = 23*a(3) + a(2) = 23*12236 + 531 = ((23 + sqrt(533))/2)^4 + ((23 - sqrt(533))/2)^4 = 281958.999996453 + 0.000003546 = 281959.
%p seq(simplify(2*(-I)^n*ChebyshevT(n, 23*I/2)), n = 0..20); # _G. C. Greubel_, Dec 29 2019
%t LinearRecurrence[{23,1},{2,23},20] (* _Harvey P. Dale_, Jul 11 2014 *)
%t LucasL[Range[20]-1,23] (* _G. C. Greubel_, Dec 29 2019 *)
%o (PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 23*I/2) ) \\ _G. C. Greubel_, Dec 29 2019
%o (Magma) I:=[2,23]; [n le 2 select I[n] else 23*Self(n-1) +Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 29 2019
%o (Sage) [2*(-I)^n*chebyshev_T(n, 23*I/2) for n in (0..20)] # _G. C. Greubel_, Dec 29 2019
%o (GAP) a:=[2,23];; for n in [3..20] do a[n]:=23*a[n-1]+a[n-2]; od; a; # _G. C. Greubel_, Dec 29 2019
%Y Cf. A089141, A051502.
%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), this sequence (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
%E Terms a(16) onward added by _G. C. Greubel_, Dec 29 2019
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