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%I M3120 N1265 #56 Sep 08 2022 08:44:29
%S 3,29,322,3571,39603,439204,4870847,54018521,599074578,6643838879,
%T 73681302247,817138163596,9062201101803,100501350283429,
%U 1114577054219522,12360848946698171,137083915467899403,1520283919093591604,16860207025497407047,186982561199565069121
%N a(n) = Lucas(5*n+2).
%C Related to Bernoulli numbers.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 141.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001947/b001947.txt">Table of n, a(n) for n = 0..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11, 1).
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F G.f.: (3 - 4*x) / (1 - 11*x - x^2). - Corrected by _Colin Barker_, Apr 22 2017
%F a(n) = Lucas(5*n+2). - _Thomas Baruchel_, Nov 26 2003
%F From _Colin Barker_, Apr 22 2017: (Start)
%F a(n) = (((11-5*sqrt(5))/2)^n*(-5+3*sqrt(5)) + (5+3*sqrt(5))*((11+5*sqrt(5))/2)^n) / (2*sqrt(5)).
%F a(n) = 11*a(n-1) + a(n-2) for n>1.
%F (End)
%p A001947:=(-3+4*z)/(-1+11*z+z**2); # Conjectured by _Simon Plouffe_ in his 1992 dissertation.
%t LucasL[5*Range[0,20]+2] (* _Harvey P. Dale_, Jan 18 2012 *)
%o (Magma) [ Lucas(5*n +2): n in [0..120]]; // _Vincenzo Librandi_, Apr 16 2011
%o (PARI) Vec((3 - 4*x) / (1 - 11*x - x^2) + O(x^20)) \\ _Colin Barker_, Apr 22 2017
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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