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A090305 a(n) = 16a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16. 1
2, 16, 258, 4144, 66562, 1069136, 17172738, 275832944, 4430499842, 71163830416, 1143051786498, 18359992414384, 294902930416642, 4736806879080656, 76083812995707138, 1222077814810394864, 19629328849962024962 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n+1)/a(n) converges to (8+sqrt(65)) = 16.0622577... Lim a(n)/a(n+1) as n approaches infinity = 0.0622577... = 1/(8+sqrt(65)) = (sqrt(65)-8). Lim a(n+1)/a(n) as n approaches infinity = 16.0622577... = (8+sqrt(65)) = 1/(sqrt(65)-8).

LINKS

Table of n, a(n) for n=0..16.

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).

FORMULA

a(n) =16a(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.: (2-16x)/(1-16x-x^2). [From Philippe Deléham, Nov 02 2008]

EXAMPLE

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

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 *)

CROSSREFS

Cf. A002070, A015910.

Sequence in context: A108242 A140307 A114039 * A246739 A283685 A197458

Adjacent sequences:  A090302 A090303 A090304 * A090306 A090307 A090308

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

More terms from Ray Chandler, Feb 14 2004

STATUS

approved

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Last modified April 30 12:33 EDT 2017. Contains 285668 sequences.