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%I #38 Jan 17 2023 18:29:31
%S 2,6,38,234,1442,8886,54758,337434,2079362,12813606,78960998,
%T 486579594,2998438562,18477210966,113861704358,701647437114,
%U 4323746327042,26644125399366,164188498723238,1011775117738794
%N a(n) = 6*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=6.
%C a(n+1)/a(n) converges to 3 + sqrt 10.
%C a(0)/a(1) = 1/3 = [3]; a(1)/a(2) = 6/38 = [6,3]; a(2)/a(3) = 38/234 = [6,6,3], a(3)/a(4) = 234/1442 = [6,6,6,3]; a(4)/a(5) = 1442/8886 = [6,6,6,6,3];...etc. Lim a(n)/a(n+1) as n approaches infinity = 0.1622776...= 1/(3 + sqrt10) = sqrt(10) - 3.
%H Vincenzo Librandi, <a href="/A085447/b085447.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Bhadouria, D. Jhala, B. Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{6,n}
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,1).
%F a(n) = (3 + sqrt 10)^n + (3 - sqrt 10)^n = A005668(n+1) + A005668(n-1).
%F O.g.f.: 2*(-1+3*x)/(-1+6*x+x^2). - _R. J. Mathar_, Dec 02 2007
%F a(n) = 2*A005667(n). - _R. J. Mathar_, Nov 10 2013
%e a(4) = 6*a(3)+a(2) = 6*234+38 = 1442.
%t RecurrenceTable[{a[0] == 2, a[1] == 6, a[n] == 6 a[n-1] + a[n-2]}, a, {n, 30}] (* _Vincenzo Librandi_, Sep 19 2016 *)
%t LinearRecurrence[{6,1}, {2,6}, 30] (* _G. C. Greubel_, Nov 07 2018 *)
%o (Magma) I:=[2,6]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 19 2016
%o (PARI) x='x+O('x^30); Vec(2*(1-3*x)/(1-6*x-x^2)) \\ _G. C. Greubel_, Nov 07 2018
%Y Cf. A005667, A005668.
%K nonn,easy
%O 0,1
%A _Gary W. Adamson_, Jul 01 2003
%E Edited and extended by _Henry Bottomley_, Jul 13 2003
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