A000962 The convergent sequence A_n for the ternary continued fraction (3,1;2,2) of period 2.

1, 0, 0, 1, 2, 5, 15, 32, 99, 210, 650, 1379, 4268, 9055, 28025, 59458, 184021, 390420, 1208340, 2563621, 7934342, 16833545, 52099395, 110534372, 342101079, 725803590, 2246343710, 4765855559, 14750202128, 31294112515, 96854484845, 205487024518, 635977131241

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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)
• D. N. Lehmer, On ternary continued fractions, Tohoku Math. J., 37 (1933), 436-445.
• 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. A000963, A000964.

Programs

Maple

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
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

Mathematica

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 *)
LinearRecurrence[{0,7,0,-3,0,1},{1,0,0,1,2,5},40] (* Harvey P. Dale, Jun 28 2020 *)

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

Formula

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