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A089772
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a(n) = Lucas(11*n).
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10
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2, 199, 39603, 7881196, 1568397607, 312119004989, 62113250390418, 12360848946698171, 2459871053643326447, 489526700523968661124, 97418273275323406890123, 19386725908489881939795601, 3858055874062761829426214722
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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, ....
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
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EXAMPLE
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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...
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MAPLE
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seq(simplify(2*(-I)^n*ChebyshevT(n, 199*I/2)), n = 0..20); # G. C. Greubel, Dec 31 2019
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MATHEMATICA
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LucasL[11*Range[0, 20]] (* or *) LinearRecurrence[{199, 1}, {2, 199}, 20] (* Harvey P. Dale, Dec 23 2015 *)
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PROG
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(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
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CROSSREFS
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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).
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KEYWORD
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easy,nonn
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AUTHOR
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Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 09 2004
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STATUS
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approved
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