Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M1473 N0582 #48 Apr 13 2022 13:25:15
%S 1,0,0,1,2,5,15,32,99,210,650,1379,4268,9055,28025,59458,184021,
%T 390420,1208340,2563621,7934342,16833545,52099395,110534372,342101079,
%U 725803590,2246343710,4765855559,14750202128,31294112515,96854484845,205487024518,635977131241
%N The convergent sequence A_n for the ternary continued fraction (3,1;2,2) of period 2.
%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 Vincenzo Librandi, <a href="/A000962/b000962.txt">Table of n, a(n) for n = 0..1000</a>
%H D. N. Lehmer, <a href="/A000962/a000962.pdf">On ternary continued fractions</a> (Annotated scanned copy)
%H D. N. Lehmer, <a href="https://www.jstage.jst.go.jp/article/tmj1911/37/0/37_0_436/_article/-char/ja/">On ternary continued fractions</a>, Tohoku Math. J., 37 (1933), 436-445.
%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_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,7,0,-3,0,1).
%F G.f.: (-2x^5 + 5x^4 + x^3 - 7x^2 + 1)/(-x^6 + 3x^4 - 7x^2 + 1).
%p A000962:=(z+1)*(2*z**4-7*z**3+6*z**2+z-1)/(-1+7*z**2-3*z**4+z**6); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p a:= n-> (Matrix([[5,2,1,0,0,1]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [0, 7, 0, -3, 0, 1][i] else 0 fi)^n)[1,6]: seq(a(n), n=0..35); # _Alois P. Heinz_, Aug 26 2008
%t CoefficientList[Series[(-2x^5+5x^4+x^3-7x^2+1)/(-x^6+3x^4-7x^2+1),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 10 2012 *)
%t LinearRecurrence[{0,7,0,-3,0,1},{1,0,0,1,2,5},40] (* _Harvey P. Dale_, Jun 28 2020 *)
%o (PARI) Vec((-2*x^5+5*x^4+x^3-7*x^2+1)/(-x^6+3*x^4-7*x^2+1)+O(x^99)) \\ _Charles R Greathouse IV_, Apr 10 2012
%Y Cf. A000963, A000964.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_