|
| |
|
|
A005667
|
|
Numerators of continued fraction convergents to sqrt(10).
(Formerly M3056)
|
|
26
|
|
|
|
1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, 243289797, 1499219281, 9238605483, 56930852179, 350823718557, 2161873163521, 13322062699683, 82094249361619, 505887558869397, 3117419602578001, 19210405174337403, 118379850648602419
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
a(2*n+1) with b(2*n+1):=A005668(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = -1, a(2*n) with b(2*n):=A005668(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = +1 (cf. Emerson reference).
Bisection: a(2*n) = T(n,19) =A078986(n), n >= 0 and a(2*n+1) = 3*S(2*n, 2*sqrt(10)), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. SeeA053120,resp.A049310.
The initial 1 corresponds to a denominator 0 inA005668.But according to standard conventions, a continued fraction starts with b(0) = integer part of the number, and the sequence of convergents p(n)/q(n) start with (p(0),q(0)) = (b(0),1). A fraction 1/0 has no mathematical meaning, the only justification is that initial terms p(-1) = 1, q(-1) = 0 are consistent with the recurrent relations p(n) = b(n)*p(n-1) + b(n-2) and the same for q(n). -M. F. Hasler,Nov 02 2019
|
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = 6*a(n-1) + a(n-2).
G.f.: (1-3*x)/(1-6*x-x^2).
a(n) = ((-i)^n)*T(n, 3*i) with T(n, x) Chebyshev's polynomials of the first kind (seeA053120) and i^2=-1.
E.g.f.: exp(3*x)*cosh(sqrt(10)*x).
a(n) = ((3+sqrt(10))^n + (3-sqrt(10))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k) * 10^k * 3^(n-2*k). (End)
a(n) = (-1)^n * a(-n) for all n in Z. -Michael Somos,Jul 14 2018 [This refers to the sequence extended to negative indices according to the recurrence relation, but not to the sequence as it is currently defined. -M. F. Hasler,Nov 02 2019]
a(n) = Lucas(n,6)/2, Lucas polynomial, L(n,x), evaluated at x = 6. -G. C. Greubel,Jun 06 2019
|
|
|
EXAMPLE
|
G.f. = 1 + 3*x + 19*x^2 + 117*x^3 + 721*x^4 + 4443*x^5 + 27379*x^6 +... -Michael Somos,Jul 14 2018
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
CoefficientList[Series[(1-3x)/(1-6x-x^2), {x, 0, 30}], x] (*Vincenzo Librandi,Jun 09 2013 *)
Join[{1}, Numerator[Convergents[Sqrt[10], 30]]] (* or *) LinearRecurrence[ {6, 1}, {1, 3}, 30] (*Harvey P. Dale,Aug 22 2016 *)
a[ n_]:= (-I)^n ChebyshevT[ n, 3 I]; (*Michael Somos,Jul 14 2018 *)
|
|
|
PROG
|
(Magma) I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; //Vincenzo Librandi,Jun 09 2013
(Sage) ((1-3*x)/(1-6*x-x^2)).series(x, 30).coefficients(x, sparse=False) #G. C. Greubel,Jun 06 2019
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
