A090305 a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.

2, 16, 258, 4144, 66562, 1069136, 17172738, 275832944, 4430499842, 71163830416, 1143051786498, 18359992414384, 294902930416642, 4736806879080656, 76083812995707138, 1222077814810394864, 19629328849962024962

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) = 0.0622577... = 1/(8+sqrt(65)) = (sqrt(65)-8).

Lim_{n-> infinity} a(n+1)/a(n) = 16.0622577... = (8+sqrt(65)) = 1/(sqrt(65)-8).

Examples

			a(4) = 16*a(3) + a(2) = 16*4144 + 258 = (8+sqrt(65))^4 + (8-sqrt(65))^4 = 66561.99998497... + 0.00001502... = 66562.
		

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 (16,1).

Crossrefs

Cf. A002070, A015910.

Lucas polynomials: A114525.

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), this sequence (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), A089772 (m=199).

Programs

GAP

m:=16;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 31 2019

Magma

m:=16; 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 31 2019

Maple

seq(simplify(2*(-I)^n*ChebyshevT(n, 8*I)), n = 0..20); # G. C. Greubel, Dec 31 2019

Mathematica

LinearRecurrence[{16,1},{2,16},40] (* or *) With[{c=Sqrt[65]}, Simplify/@ Table[(c-8)((8+c)^n-(8-c)^n (129+16c)),{n,20}]] (* Harvey P. Dale, Oct 31 2011 *)
LucasL[Range[20]-1, 16] (* G. C. Greubel, Dec 31 2019 *)

PARI

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 8*I) ) \\ G. C. Greubel, Dec 31 2019

Sage

[2*(-I)^n*chebyshev_T(n, 8*I) for n in (0..20)] # G. C. Greubel, Dec 31 2019

Formula

a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.

a(n) = (8+sqrt(65))^n + (8-sqrt(65))^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.: (2-16*x)/(1-16*x-x^2). - Philippe Deléham, Nov 02 2008

a(n) = Lucas(n, 16) = 2*(-i)^n * ChebyshevT(n, 8*i). - G. C. Greubel, Dec 31 2019

E.g.f.: 2*exp(8*x)*cosh(sqrt(65)*x). - Stefano Spezia, Jan 01 2020

Extensions

More terms from Ray Chandler, Feb 14 2004