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A000962
The convergent sequence A_n for the ternary continued fraction (3,1;2,2) of period 2.
(Formerly M1473 N0582)
3
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
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
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
Sequence in context: A299159 A006451 A226103 * A118387 A245961 A034522
KEYWORD
nonn,easy
STATUS
approved