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A087215 Lucas(6*n): a(n) = 18*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 18. 7
2, 18, 322, 5778, 103682, 1860498, 33385282, 599074578, 10749957122, 192900153618, 3461452808002, 62113250390418, 1114577054219522, 20000273725560978, 358890350005878082, 6440026026380244498, 115561578124838522882, 2073668380220713167378 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n+1)/a(n) converges to 9+sqrt(80) = 17.9442719... a(0)/a(1)=2/18; a(1)/a(2)=18/322; a(2)/a(3)=322/5778; a(3)/a(4)=5778/103682; ... etc.

Lim a(n)/a(n+1) as n approaches infinity = 0.05572809000084... = 1/(9+sqrt(80)) = 9-sqrt(80).

LINKS

Colin Barker, Table of n, a(n) for n = 0..750

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence R_4.

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 (18,-1).

FORMULA

a(n) = A000032(6n).

a(n) =18*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 18.

a(n) = (9+sqrt(80))^n + (9-sqrt(80))^n.

G.f.: 2*(1-9*x)/(1-18*x+x^2). - Philippe Deléham, Nov 17 2008

a(n) = 2*A023039(n). - R. J. Mathar, Oct 22 2010

EXAMPLE

a(4) = 103682 = 18*a(3) - a(2) = 18*5778 - 322 =(9+sqrt(80))^4 + (9-sqrt(80))^4 =

103681.99999035512 + 0.00000964487 = 103682.

MATHEMATICA

a[0] = 2; a[1] = 18; a[n_] := 18a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

PROG

(MAGMA) [ Lucas(6*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

(PARI) Vec(2*(1-9*x)/(1-18*x+x^2) + O(x^20)) \\ Colin Barker, Jan 24 2016

CROSSREFS

Cf. A074919.

Row 2 * 2 of array A188645.

Sequence in context: A192555 A179497 A227325 * A229490 A192985 A193264

Adjacent sequences:  A087212 A087213 A087214 * A087216 A087217 A087218

KEYWORD

easy,nonn

AUTHOR

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003

STATUS

approved

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Last modified March 23 20:32 EDT 2017. Contains 283964 sequences.