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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000962, A000964.

Programs

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 *)

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

Extensions

More terms from Emeric Deutsch, Apr 22 2006