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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 (list; graph; refs; listen; history; text; internal format)
 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. 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 STATUS approved

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Last modified April 10 14:07 EDT 2021. Contains 342845 sequences. (Running on oeis4.)