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. See A053120, resp. A049310.
The initial 1 corresponds to a denominator 0 in A005668. 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
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Tian-Xiao He and Peter J.-S. Shiue, Identities for linear recursive sequences of order 2, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
Tanya Khovanova, Recursive Sequences
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (6,1).
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 (see A053120) and i^2=-1.
From Paul Barry, Nov 15 2003: (Start)
Binomial transform of A084132.
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
A005667:=(-1+3*z)/(-1+6*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Join[{1}, Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[10], n]]], {n, 1, 30}]] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
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 *)
LucasL[Range[0, 30], 6]/2 (* G. C. Greubel, Jun 06 2019 *)
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
(PARI) a(n)=([0, 1; 1, 6]^n*[1; 3])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015
(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
Chebyshev comments from Wolfdieter Lang, Jan 10 2003
STATUS
approved