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A005667 Numerators of continued fraction convergents to sqrt(10).

%I A005667 M3056 #91 Jan 05 2025 19:51:33
%S A005667 1,3,19,117,721,4443,27379,168717,1039681,6406803,39480499,243289797,
%T A005667 1499219281,9238605483,56930852179,350823718557,2161873163521,
%U A005667 13322062699683,82094249361619,505887558869397,3117419602578001,19210405174337403,118379850648602419
%N A005667 Numerators of continued fraction convergents to sqrt(10).
%C A005667 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).
%C A005667 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.
%C A005667 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
%D A005667 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005667 Vincenzo Librandi, <a href="/A005667/b005667.txt">Table of n, a(n) for n = 0..1000</a>
%H A005667 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
%H A005667 R. K. Guy, <a href="/A005667/a005667.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A005667 Tian-Xiao He and Peter J.-S. Shiue, <a href="https://doi.org/10.3934/era.2021049">Identities for linear recursive sequences of order 2</a>, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
%H A005667 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A005667 Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and 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 A005667 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 A005667 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005667 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A005667 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,1).
%F A005667 a(n) = 6*a(n-1) + a(n-2).
%F A005667 G.f.: (1-3*x)/(1-6*x-x^2).
%F A005667 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.
%F A005667 From _Paul Barry_, Nov 15 2003: (Start)
%F A005667 Binomial transform of A084132.
%F A005667 E.g.f.: exp(3*x)*cosh(sqrt(10)*x).
%F A005667 a(n) = ((3+sqrt(10))^n + (3-sqrt(10))^n)/2.
%F A005667 a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k) * 10^k * 3^(n-2*k). (End)
%F A005667 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]
%F A005667 a(n) = Lucas(n,6)/2, Lucas polynomial, L(n,x), evaluated at x = 6. - _G. C. Greubel_, Jun 06 2019
%e A005667 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
%p A005667 A005667:=(-1+3*z)/(-1+6*z+z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A005667 Join[{1},Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[10],n]]],{n,1,30}]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%t A005667 CoefficientList[Series[(1-3x)/(1-6x-x^2), {x,0,30}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t A005667 Join[{1},Numerator[Convergents[Sqrt[10],30]]] (* or *) LinearRecurrence[ {6,1},{1,3},30] (* _Harvey P. Dale_, Aug 22 2016 *)
%t A005667 a[ n_] := (-I)^n ChebyshevT[ n, 3 I]; (* _Michael Somos_, Jul 14 2018 *)
%t A005667 LucasL[Range[0,30], 6]/2 (* _G. C. Greubel_, Jun 06 2019 *)
%o A005667 (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
%o A005667 (PARI) a(n)=([0,1;1,6]^n*[1;3])[1,1] \\ _Charles R Greathouse IV_, Jun 11 2015
%o A005667 (Sage) ((1-3*x)/(1-6*x-x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 06 2019
%Y A005667 Cf. A010467, A040006, A084134, A005668 (denominators).
%K A005667 nonn,frac,easy
%O A005667 0,2
%A A005667 _N. J. A. Sloane_, _R. K. Guy_
%E A005667 Chebyshev comments from _Wolfdieter Lang_, Jan 10 2003