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A089772 a(n) = Lucas(11*n). 10
2, 199, 39603, 7881196, 1568397607, 312119004989, 62113250390418, 12360848946698171, 2459871053643326447, 489526700523968661124, 97418273275323406890123, 19386725908489881939795601, 3858055874062761829426214722 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Lim_{n-> infinity} a(n+1)/a(n) = 199.00502499874... = (199 + sqrt(39605))/2.

Lim_{n->infinity} a(n)/a(n+1) = 0.00502499874... = 2/(199 + sqrt(39605)) = (sqrt(39605) - 199)/2.

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..100

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for linear recurrences with constant coefficients, signature (199,1).

FORMULA

a(n) = 199*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 199.

a(n) = ((199 + sqrt(39605))/2)^n + ((199 - sqrt(39605))/2)^n.

a(n)^2 = a(2n) - 2 if n = 1, 3, 5, ...;

a(n)^2 = a(2n) + 2 if n = 2, 4, 6, ....

G.f.: (2 - 199*x)/(1 - 199*x - x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 199) = 2*(-i)^n * ChebyshevT(n, 199*i/2). - G. C. Greubel, Dec 31 2019

E.g.f.: 2*exp(199*x/2)*cosh(sqrt(39605)*x/2). - Stefano Spezia, Jan 01 2020

EXAMPLE

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...

MAPLE

seq(simplify(2*(-I)^n*ChebyshevT(n, 199*I/2)), n = 0..20); # G. C. Greubel, Dec 31 2019

MATHEMATICA

LucasL[11*Range[0, 20]] (* or *) LinearRecurrence[{199, 1}, {2, 199}, 20] (* Harvey P. Dale, Dec 23 2015 *)

LucasL[Range[20]-1, 199] (* G. C. Greubel, Dec 31 2019 *)

PROG

(MAGMA) [Lucas(11*n): n in [0..20]]; // Vincenzo Librandi, Apr 15 2011

(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 199*I/2) ) \\ G. C. Greubel, Dec 31 2019

(Sage) [lucas_number2(11*n, 1, -1) for n in (0..20)] # G. C. Greubel, Dec 30 2019

(GAP) List([0..20], n-> Lucas(1, -1, 11*n)[2] ); # G. C. Greubel, Dec 30 2019

CROSSREFS

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).

Sequence in context: A033147 A213162 A124339 * A280090 A195742 A180301

Adjacent sequences:  A089769 A089770 A089771 * A089773 A089774 A089775

KEYWORD

easy,nonn

AUTHOR

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 09 2004

STATUS

approved

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Last modified September 21 13:00 EDT 2020. Contains 337272 sequences. (Running on oeis4.)