A090308 a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.

2, 19, 363, 6916, 131767, 2510489, 47831058, 911300591, 17362542287, 330799604044, 6302555019123, 120079344967381, 2287810109399362, 43588471423555259, 830468767156949283, 15822495047405591636

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) = 0.052486... = 2/(19+sqrt(365)) = (sqrt(365)-19)/2.

Lim_{n-> infinity} a(n+1)/a(n) = 19.052486... = (19+sqrt(365))/2 = 2/(sqrt(365)-19).

Examples

			a(4) = 19*a(3) + a(2) = 19*6916 + 363 = ((19+sqrt(365))/2)^4 + ((19-sqrt(365))/2)^4 = 131766.9999924108 + 0.0000075891 = 131767.
		

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 (19,1).

Crossrefs

Cf. A049270.

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), this sequence (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

Programs

GAP

m:=19;; 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:=19; 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, 19*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

Mathematica

LucasL[Range[20]-1,20] (* G. C. Greubel, Dec 30 2019 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 19*I/2) ) \\ G. C. Greubel, Dec 30 2019

Sage

[2*(-I)^n*chebyshev_T(n, 19*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

Formula

a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.

a(n) = ((19+sqrt(365))/2)^n + ((19-sqrt(365))/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.: (2-19*x)/(1-19*x-x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 19) = 2*(-i)^n * ChebyshevT(n, 19*i/2). - G. C. Greubel, Dec 30 2019

Extensions

More terms from Ray Chandler, Feb 14 2004