A000963 The convergent sequence B_n for the ternary continued fraction (3,1;2,2) of period 2. (Formerly M2660 N1062)

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

0,4

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).

Formula

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

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 *)

Crossrefs

Cf. A000962, A000964.

Sequence in context: A341576 A143817 A297210 * A364646 A133593 A297154

Adjacent sequences: A000960 A000961 A000962 * A000964 A000965 A000966

Keyword

nonn,cofr,easy

Author

N. J. A. Sloane

Status

approved