%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_