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A000963 The convergent sequence B_n for the ternary continued fraction (3,1;2,2) of period 2.

%I A000963 M2660 N1062 #36 Apr 13 2022 13:25:15
%S A000963 0,1,0,3,7,16,49,104,322,683,2114,4485,13881,29450,91147,193378,
%T A000963 598500,1269781,3929940,8337783,25805227,54748516,169445269,359496044,
%U A000963 1112631142
%N A000963 The convergent sequence B_n for the ternary continued fraction (3,1;2,2) of period 2.
%D A000963 D. N. Lehmer, On ternary continued fractions, Tohoku Math. J., 37 (1933), 436-445.
%D A000963 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000963 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000963 Vincenzo Librandi, <a href="/A000963/b000963.txt">Table of n, a(n) for n = 0..1000</a>
%H A000963 D. N. Lehmer, <a href="/A000962/a000962.pdf">On ternary continued fractions</a> (Annotated scanned copy)
%H A000963 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 A000963 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000963 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,7,0,-3,0,1).
%F A000963 G.f.: (-2x^5 + 7x^4 - 4x^3 + x)/(-x^6 + 3x^4 - 7x^2 + 1).
%p A000963 A000963:=z*(-1+4*z**2-7*z**3+2*z**4)/(-1+7*z**2-3*z**4+z**6); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p A000963 a:= n-> (Matrix([[16,7,3,0,1,0]]). 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..24); # _Alois P. Heinz_, Aug 26 2008
%t A000963 CoefficientList[Series[(-2x^5+7x^4-4x^3+x)/(-x^6+3x^4-7x^2+1),{x,0,40}],x] (* _Vincenzo Librandi_, Apr 11 2012 *)
%t A000963 LinearRecurrence[{0,7,0,-3,0,1},{0,1,0,3,7,16},30] (* _Harvey P. Dale_, Sep 06 2021 *)
%Y A000963 Cf. A000962, A000964.
%K A000963 nonn,cofr,easy
%O A000963 0,4
%A A000963 _N. J. A. Sloane_