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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_{n -> inf} a(n)/a(n+1) = 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 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 *) Table[LucasL[6n], {n, 0, 18}]  (* or *) CoefficientList[Series[2*(1 - 9*x)/(1 - 18*x + x^2), {x, 0, 17}], x] (* Indranil Ghosh, Mar 15 2017 *) 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 (PARI) a(n) = if(n<2, 17^n + 1, 18*a(n - 1) - a(n - 2)); for(n=0, 17, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 15 2017 CROSSREFS Cf. A074919. Row 2 * 2 of array A188645. Sequence in context: A179497 A296837 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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