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A090309
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a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.
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14
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2, 20, 402, 8060, 161602, 3240100, 64963602, 1302512140, 26115206402, 523606640180, 10498248010002, 210488566840220, 4220269584814402, 84615880263128260, 1696537874847379602, 34015373377210720300
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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)/a(n+1) = 0.0498756... = 1/(10+sqrt(101)) = (sqrt(101)-10).
Lim_{n-> infinity} a(n+1)/a(n) = 20.0498756... = (10+sqrt(101)) = 1/(sqrt(101)-10).
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LINKS
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FORMULA
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a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.
a(n) = (10 + sqrt(101))^n + (10 - sqrt(101))^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, 20) = 2*(-i)^n * ChebyshevT(n, 10*i). - G. C. Greubel, Dec 30 2019
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EXAMPLE
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a(4) = 20*a(3) + a(2) = 20*8060 + 402 = (10+sqrt(101))^4 + (10-sqrt(101))^4 = 161601.999993811 + 0.000006188 = 161602.
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MAPLE
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seq(simplify(2*(-I)^n*ChebyshevT(n, 10*I)), n = 0..20); # G. C. Greubel, Dec 30 2019
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MATHEMATICA
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LinearRecurrence[{20, 1}, {2, 20}, 20] (* Harvey P. Dale, Nov 19 2015 *)
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PROG
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(PARI) vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 10*I) ) \\ G. C. Greubel, Dec 30 2019
(Magma) m:=20; 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
(Sage) [2*(-I)^n*chebyshev_T(n, 10*I) for n in (0..20)] # G. C. Greubel, Dec 30 2019
(GAP) m:=20;; 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
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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), this sequence (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).
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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 25 2004
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EXTENSIONS
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STATUS
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approved
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