A001081 - OEIS

%I M4573 N1949 #108 Jun 14 2024 09:41:34

%S 1,8,127,2024,32257,514088,8193151,130576328,2081028097,33165873224,

%T 528572943487,8424001222568,134255446617601,2139663144659048,

%U 34100354867927167,543466014742175624,8661355881006882817

%N a(n) = 16*a(n-1) - a(n-2).

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

%C 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.

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

%C 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

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

%C a(n+3)*a(n) - a(n+2)*a(n+1) = 16*63. - _Bruno Berselli_, Feb 18 2013

%D 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

%H T. D. Noe, <a href="/A001081/b001081.txt">Table of n, a(n) for n = 0..200</a>

%H H. Brocard, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN598948236_0004/DMDLOG_0053">Notes élémentaires sur le problème de Peel [sic]</a>, Nouvelle Correspondance Mathématique, 4 (1878), 337-343.

%H M. Davis, <a href="http://www.rand.org/pubs/research_memoranda/RM5494.html">One equation to rule them all</a>, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, <a href="https://arxiv.org/abs/1904.13002">The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d))</a>, arXiv:1904.13002 [math.NT], 2019.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H Mihai Prunescu, <a href="https://arxiv.org/abs/2406.06436">On other two representations of the C-recursive integer sequences by terms in modular arithmetic</a>, arXiv:2406.06436 [math.NT], 2024. See p. 17.

%H Mihai Prunescu and Lorenzo Sauras-Altuzarra, <a href="https://arxiv.org/abs/2405.04083">On the representation of C-recursive integer sequences by arithmetic terms</a>, arXiv:2405.04083 [math.LO], 2024. See p. 16.

%H N. J. Wildberger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Wildberger/wildberger2.html">Pell's equation without irrational numbers</a>, J. Int. Seq. 13 (2010), 10.4.3, Section 5.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).

%F G.f.: (1-8*x)/(1-16*x+x^2). - _Simon Plouffe_ in his 1992 dissertation.

%F 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

%F 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).

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

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

%F a(n) = 16*a(n-1) - a(n-2) with a(1)=1 and a(2)=8. - _Sture Sjöstedt_, Nov 18 2011

%F a(n) = A077412(n) - 8*A077412(n-1). - _R. J. Mathar_, Jul 22 2017

%F a(n) = (-i)^n*Lucas(n, 16*i)/2, where i = sqrt(-1). - _G. C. Greubel_, Jun 06 2019

%t LinearRecurrence[{16, -1}, {1, 8}, 30]

%t CoefficientList[Series[(1-8*x)/(1-16*x+x^2), {x, 0, 30}], x] (* _G. C. Greubel_, Dec 20 2017 *)

%t Table[LucasL[n, 16*I]*(-I)^n/2, {n,0,30}] (* _G. C. Greubel_, Jun 06 2019 *)

%o (Sage) [lucas_number2(n,16,1)/2 for n in range(0,30)] # _Zerinvary Lajos_, Jun 26 2008

%o (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

%o (PARI) Vec((1-8*x)/(1-16*x+x^2)+O(x^30)) \\ _Charles R Greathouse IV_, Jul 02 2013

%Y Cf. A090727, A001080, A010727.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Chebyshev and Pell comments from _Wolfdieter Lang_, Nov 08 2002