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A000962
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The convergent sequence A_n for the ternary continued fraction (3,1;2,2) of period 2.
(Formerly M1473 N0582)
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3
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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
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OFFSET
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0,5
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REFERENCES
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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).
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LINKS
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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).
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FORMULA
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G.f.: (-2x^5 + 5x^4 + x^3 - 7x^2 + 1)/(-x^6 + 3x^4 - 7x^2 + 1).
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000963, A000964.
Sequence in context: A299159 A006451 A226103 * A118387 A245961 A034522
Adjacent sequences: A000959 A000960 A000961 * A000963 A000964 A000965
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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