A000962 The convergent sequence A_n for the ternary continued fraction (3,1;2,2) of period 2. (Formerly M1473 N0582)

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

0,5

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).

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A000963, A000964.

Sequence in context: A299159 A006451 A226103 * A118387 A245961 A034522

Adjacent sequences: A000959 A000960 A000961 * A000963 A000964 A000965

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved