%I #21 Sep 08 2022 08:45:12
%S 2,15,227,3420,51527,776325,11696402,176222355,2655031727,40001698260,
%T 602680505627,9080209282665,136805819745602,2061167505466695,
%U 31054318401746027,467875943531657100,7049193471376602527
%N a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.
%C Lim_{n-> infinity} a(n)/a(n+1) = 0.066372... = 2/(15+sqrt(229)) = (sqrt(229)-15)/2.
%C Lim_{n-> infinity} a(n+1)/a(n) = 15.066372... = (15+sqrt(229))/2 = 2/(sqrt(229)-15).
%C For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - _Johannes W. Meijer_, Jun 12 2010
%H G. C. Greubel, <a href="/A090301/b090301.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 (15,1).
%F a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.
%F a(n) = ((15+sqrt(229))/2)^n + ((15-sqrt(229))/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.: (2-15*x)/(1-15*x-x^2). - _Philippe Deléham_, Nov 02 2008
%F Contribution from _Johannes W. Meijer_, Jun 12 2010: (Start)
%F Lim_{k-> infinity} a(n+k)/a(k) = (A090301(n) + A154597(n)*sqrt(229))/2.
%F Lim_{n-> infinity} A090301(n)/ A154597(n) = sqrt(229).
%F a(2n+1) = 15*A098246(n).
%F a(3n+1) = A041426(5n), a(3n+2) = A041426(5n+3), a(3n+3) = 2*A041426(5n+4).
%F (End)
%F a(n) = Lucas(n, 15) = 2*(-i)^n * ChebyshevT(n, 15*i/2). - _G. C. Greubel_, Dec 31 2019
%F E.g.f.: 2*exp(15*x/2)*cosh(sqrt(229)*x/2). - _Stefano Spezia_, Jan 01 2020
%e a(4) = 15*a(3) + a(2) = 15*3420 + 227 = ((15+sqrt(229))/2)^4 + ((15-sqrt(229))/2)^4 = 51526.9999805 + 0.0000194 = 51527.
%p seq(simplify(2*(-I)^n*ChebyshevT(n, 15*I/2)), n = 0..20); # _G. C. Greubel_, Dec 31 2019
%t LucasL[Range[20]-1, 15] (* _G. C. Greubel_, Dec 31 2019 *)
%o (PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 15*I/2) ) \\ _G. C. Greubel_, Dec 3012019
%o (Magma) m:=15; 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 31 2019
%o (Sage) [2*(-I)^n*chebyshev_T(n, 15*I/2) for n in (0..20)] # _G. C. Greubel_, Dec 31 2019
%o (GAP) m:=15;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # _G. C. Greubel_, Dec 31 2019
%Y Cf. A058087, A071416.
%Y Lucas polynomials: A114525.
%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), this sequence (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), A089772 (m=199).
%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