A090309 a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.

2, 20, 402, 8060, 161602, 3240100, 64963602, 1302512140, 26115206402, 523606640180, 10498248010002, 210488566840220, 4220269584814402, 84615880263128260, 1696537874847379602, 34015373377210720300

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Keywords

• easy
• nonn

Comments

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).

Examples

			a(4) = 20*a(3) + a(2) = 20*8060 + 402 = (10+sqrt(101))^4 + (10-sqrt(101))^4 = 161601.999993811 + 0.000006188 = 161602.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• 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 (20,1).

Crossrefs

Cf. A002116.

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).

Programs

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

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

Maple

seq(simplify(2*(-I)^n*ChebyshevT(n, 10*I)), n = 0..20); # G. C. Greubel, Dec 30 2019

Mathematica

LinearRecurrence[{20,1},{2,20},20] (* Harvey P. Dale, Nov 19 2015 *)
LucasL[Range[20]-1,20] (* G. C. Greubel, Dec 30 2019 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 10*I) ) \\ 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

Formula

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, ... .

G.f.: (2-20*x)/(1-20*x-x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 20) = 2*(-i)^n * ChebyshevT(n, 10*i). - G. C. Greubel, Dec 30 2019

Extensions

More terms from Ray Chandler, Feb 14 2004