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

%I A001081 M4573 N1949 #111 Feb 28 2025 09:40:11
%S A001081 1,8,127,2024,32257,514088,8193151,130576328,2081028097,33165873224,
%T A001081 528572943487,8424001222568,134255446617601,2139663144659048,
%U A001081 34100354867927167,543466014742175624,8661355881006882817
%N A001081 a(n) = 16*a(n-1) - a(n-2).
%C A001081 Chebyshev's polynomials T(n,x) evaluated at x=8.
%C A001081 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 A001081 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 A001081 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 A001081 a(n)^2 - 7 * A001080(n)^2 = 1 (this property is equivalent to the second comment). - _Vincenzo Librandi_, Feb 17 2013
%C A001081 a(n+3)*a(n) - a(n+2)*a(n+1) = 16*63. - _Bruno Berselli_, Feb 18 2013
%D A001081 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 A001081 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001081 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001081 V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.
%H A001081 T. D. Noe, <a href="/A001081/b001081.txt">Table of n, a(n) for n = 0..200</a>
%H A001081 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 A001081 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 A001081 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A001081 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 A001081 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 A001081 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001081 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 A001081 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 A001081 Mihai Prunescu and Joseph M. Shunia, <a href="https://arxiv.org/abs/2502.16928">On modular representations of C-recursive integer sequences</a>, arXiv:2502.16928 [math.NT], 2025. See p. 5.
%H A001081 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 A001081 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001081 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F A001081 G.f.: (1-8*x)/(1-16*x+x^2). - _Simon Plouffe_ in his 1992 dissertation.
%F A001081 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 A001081 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 A001081 a(n) = ((8 + 3*sqrt(7))^n + (8 - 3*sqrt(7))^n)/2.
%F A001081 a(n) = sqrt(63*A077412(n-1)^2 + 1), n>=1, (cf. Richardson comment).
%F A001081 a(n) = 16*a(n-1) - a(n-2) with a(1)=1 and a(2)=8. - _Sture Sjöstedt_, Nov 18 2011
%F A001081 a(n) = A077412(n) - 8*A077412(n-1). - _R. J. Mathar_, Jul 22 2017
%F A001081 a(n) = (-i)^n*Lucas(n, 16*i)/2, where i = sqrt(-1). - _G. C. Greubel_, Jun 06 2019
%t A001081 LinearRecurrence[{16, -1}, {1, 8}, 30]
%t A001081 CoefficientList[Series[(1-8*x)/(1-16*x+x^2), {x, 0, 30}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%t A001081 Table[LucasL[n, 16*I]*(-I)^n/2, {n,0,30}] (* _G. C. Greubel_, Jun 06 2019 *)
%o A001081 (Sage) [lucas_number2(n,16,1)/2 for n in range(0,30)] # _Zerinvary Lajos_, Jun 26 2008
%o A001081 (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 A001081 (PARI) Vec((1-8*x)/(1-16*x+x^2)+O(x^30)) \\ _Charles R Greathouse IV_, Jul 02 2013
%Y A001081 Cf. A090727, A001080, A010727.
%K A001081 nonn,easy
%O A001081 0,2
%A A001081 _N. J. A. Sloane_
%E A001081 Chebyshev and Pell comments from _Wolfdieter Lang_, Nov 08 2002