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

0, 1, 0, 3, 7, 16, 49, 104, 322, 683, 2114, 4485, 13881, 29450, 91147, 193378, 598500, 1269781, 3929940, 8337783, 25805227, 54748516, 169445269, 359496044, 1112631142

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy

References

• D. N. Lehmer, On ternary continued fractions, Tohoku Math. J., 37 (1933), 436-445.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. N. Lehmer, On ternary continued fractions (Annotated scanned copy)
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (0,7,0,-3,0,1).

Crossrefs

Cf. A000962, A000964.

Programs

Maple

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
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

Mathematica

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 *)
LinearRecurrence[{0,7,0,-3,0,1},{0,1,0,3,7,16},30] (* Harvey P. Dale, Sep 06 2021 *)

Formula

G.f.: (-2x^5 + 7x^4 - 4x^3 + x)/(-x^6 + 3x^4 - 7x^2 + 1).