A001081 a(n) = 16*a(n-1) - a(n-2). (Formerly M4573 N1949)

1, 8, 127, 2024, 32257, 514088, 8193151, 130576328, 2081028097, 33165873224, 528572943487, 8424001222568, 134255446617601, 2139663144659048, 34100354867927167, 543466014742175624, 8661355881006882817

Offset

0,2

Comments

Chebyshev's polynomials T(n,x) evaluated at x=8.

The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 63*b(n)^2 = +1 with b(n)= A077412(n-1), n>=1 and b(0)=0.

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(7). - Benoit Cloitre, Feb 14 2004

a(7+14k)-1 and a(7+14k)+1 are consecutive odd powerful numbers. The first pair is 130576328+-1. See A076445. - T. D. Noe, May 04 2006

a(n)^2 - 7 * A001080(n)^2 = 1 (this property is equivalent to the second comment). - Vincenzo Librandi, Feb 17 2013

a(n+3)*a(n) - a(n+2)*a(n+1) = 16*63. - Bruno Berselli, Feb 18 2013

References

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012

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).

V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

Links

T. D. Noe, Table of n, a(n) for n = 0..200

H. Brocard, Notes élémentaires sur le problème de Peel [sic], Nouvelle Correspondance Mathématique, 4 (1878), 337-343.

M. Davis, One equation to rule them all, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773.

Tanya Khovanova, Recursive Sequences

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, 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

N. J. Wildberger, Pell's equation without irrational numbers, J. Int. Seq. 13 (2010), 10.4.3, Section 5.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (16,-1).

Formula

G.f.: (1-8*x)/(1-16*x+x^2). - Simon Plouffe in his 1992 dissertation.

For all members x of the sequence, 7*x^2 - 7 is a square. Limit_{n->infinity} a(n)/a(n-1) = 8 + 3*sqrt(7). - Gregory V. Richardson, Oct 13 2002

a(n) = T(n, 8) = (S(n, 16)-S(n-2, 16))/2, with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 16)= A077412(n).

a(n) = ((8 + 3*sqrt(7))^n + (8 - 3*sqrt(7))^n)/2.

a(n) = sqrt(63*A077412(n-1)^2 + 1), n>=1, (cf. Richardson comment).

a(n) = 16*a(n-1) - a(n-2) with a(1)=1 and a(2)=8. - Sture Sjöstedt, Nov 18 2011

a(n) = A077412(n) - 8*A077412(n-1). - R. J. Mathar, Jul 22 2017

a(n) = (-i)^n*Lucas(n, 16*i)/2, where i = sqrt(-1). - G. C. Greubel, Jun 06 2019

Mathematica

LinearRecurrence[{16, -1}, {1, 8}, 30]
CoefficientList[Series[(1-8*x)/(1-16*x+x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 20 2017 *)
Table[LucasL[n, 16*I]*(-I)^n/2, {n, 0, 30}] (* G. C. Greubel, Jun 06 2019 *)

Prog

(Sage) [lucas_number2(n, 16, 1)/2 for n in range(0, 30)] # Zerinvary Lajos, Jun 26 2008
(Magma) I:=[1, 8]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 17 2013
(PARI) Vec((1-8*x)/(1-16*x+x^2)+O(x^30)) \\ Charles R Greathouse IV, Jul 02 2013

Crossrefs

Cf. A090727, A001080, A010727.

Sequence in context: A220728 A355766 A029472 * A371368 A178790 A297214

Adjacent sequences: A001078 A001079 A001080 * A001082 A001083 A001084

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002

Status

approved