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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.
• Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 17.
• Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
• Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 5.
• 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).

Crossrefs

Cf. A090727, A001080, A010727.

Programs

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

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 *)

PARI

Vec((1-8*x)/(1-16*x+x^2)+O(x^30)) \\ Charles R Greathouse IV, Jul 02 2013

Sage

[lucas_number2(n,16,1)/2 for n in range(0,30)] # Zerinvary Lajos, Jun 26 2008

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

Extensions

Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002