A089775 Lucas numbers L(12n).

2, 322, 103682, 33385282, 10749957122, 3461452808002, 1114577054219522, 358890350005878082, 115561578124838522882, 37210469265847998489922, 11981655542024930675232002, 3858055874062761829426214722, 1242282009792667284144565908482, 400010949097364802732720796316482

Offset

0,1

Comments

a(n+1)/a(n) converges to (322 + sqrt(103680))/2 = 321.996894379... a(0)/a(1) = 2/322; a(1)/a(2) = 322/103682; a(2)/a(3) = 103682/33385282; a(3)/a(4) = 33385282/10749957122; etc. Lim_{n -> inf} a(n)/a(n+1) = 0.00310562... = 2/(322 + sqrt(103680)) = (322 - sqrt(103680))/2.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..100

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 (322, -1).

Formula

a(n) = 322*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 322

a(n) = ((322 + sqrt(103680))/2)^n + ((322 - sqrt(103680))/2)^n.

(a(n))^2 = a(2n) + 2.

G.f.: (2-322*x)/(1-322*x+x^2). - Philippe Deléham, Nov 02 2008

Example

a(4) = 10749957122 = 322*a(3) - a(2) = 322*33385282 - 103682 = ((322 + sqrt(103680))/2)^4 + ((322 - sqrt(103680))/2)^4.

Mathematica

Table[LucasL[12n], {n, 0, 13}] (* Indranil Ghosh, Mar 15 2017 *)

Prog

(Magma) [ Lucas(12*n) : n in [0..70]]; // Vincenzo Librandi, Apr 15 2011
(PARI) Vec((2 - 322*x)/(1 - 322*x + x^2) + O(x^14)) \\ Indranil Ghosh, Mar 15 2017

Crossrefs

Cf. A000032, A060964.

a(n) = A000032(12n).

Row 9 * 2 of array A188644

Sequence in context: A221190 A367927 A192725 * A094402 A332132 A262637

Adjacent sequences: A089772 A089773 A089774 * A089776 A089777 A089778

Keyword

easy,nonn

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 09 2004

Extensions

a(11) - a(13) from Vincenzo Librandi, Apr 15 2011

Status

approved