A000964 The convergent sequence C_n for the ternary continued fraction (3,1;2,2) of period 2. (Formerly M3343 N1345)

0, 0, 1, 1, 4, 8, 25, 53, 164, 348, 1077, 2285, 7072, 15004, 46437, 98521, 304920, 646920, 2002201, 4247881, 13147084, 27892928, 86327905, 183153773, 566856284, 1202645508, 3722157357, 7896950165, 24440860552, 51853868404, 160486408077

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, Tohoku Math. J., 37 (1933), 436-445.

D. N. Lehmer, On ternary continued fractions (Annotated scanned copy)

Index entries for linear recurrences with constant coefficients, signature (0, 7, 0, -3, 0, 1).

Formula

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

a(n) = 7*a(n-2) - 3*a(n-4) + a(n-6); a(0)=0, a(1)=0, a(2)=1, a(3)=1, a(4)=4, a(5)=8. - Harvey P. Dale, Jun 29 2011

Maple

G:=(x^5-3*x^4+x^3+x^2)/(-x^6+3*x^4-7*x^2+1): Gser:=series(G, x=0, 35): seq(coeff(Gser, x, n), n=0..32); # Emeric Deutsch, Apr 22 2006

Mathematica

LinearRecurrence[{0, 7, 0, -3, 0, 1}, {0, 0, 1, 1, 4, 8}, 31] (* Harvey P. Dale, Jun 29 2011 *)
CoefficientList[Series[(x^5-3x^4+x^3+x^2)/(-x^6+3x^4-7x^2+1), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 11 2012 *)

Crossrefs

Cf. A000962, A000964.

Sequence in context: A185615 A068367 A292548 * A297458 A328038 A107840

Adjacent sequences: A000961 A000962 A000963 * A000965 A000966 A000967

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Apr 22 2006

Status

approved