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A087265 Lucas numbers L(8*n). 3
2, 47, 2207, 103682, 4870847, 228826127, 10749957122, 505019158607, 23725150497407, 1114577054219522, 52361396397820127, 2459871053643326447, 115561578124838522882, 5428934300813767249007, 255044350560122222180447 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n+1)/a(n) converges to (47+sqrt(2205))/2 = 46.9787137... a(0)/a(1)=2/47; a(1)/a(2)=47/2207; a(2)/a(3)=2207/103682; a(3)/a(4)=103682/4870847; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0,02128623625... = 2/(47+sqrt(2205)) = (47-sqrt(2205))/2.

a(n) = a(-n). - Alois P. Heinz, Aug 07 2008

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..596

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 (47, -1).

FORMULA

a(n) = 47*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 47.

a(n) = ((47+sqrt(2205))/2)^n + ((47-sqrt(2205))/2)^n

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

G.f.: (2-47*x)/(1-47*x+x^2). - Alois P. Heinz, Aug 07 2008

EXAMPLE

a(4) = 4870847 = 47*a(3) - a(2) = 47*103682 - 2207=((47+sqrt(2205))/2)^4 + ( (47-sqrt(2205))/2)^4 =4870846.999999794696 + 0.000000205303 = 4870847.

MAPLE

a:= n-> (Matrix([[2, 47]]). Matrix([[47, 1], [ -1, 0]])^(n))[1, 1]:

seq(a(n), n=0..14);  # Alois P. Heinz, Aug 07 2008

PROG

(MAGMA) [ Lucas(8*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

CROSSREFS

Cf. A000032.

a(n) = A000032(8n).

Sequence in context: A277655 A246543 A119776 * A079307 A005814 A177190

Adjacent sequences:  A087262 A087263 A087264 * A087266 A087267 A087268

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

Terms a(22)-a(27) from John W. Layman, Jun 14 2004

STATUS

approved

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Last modified July 25 14:33 EDT 2017. Contains 289795 sequences.