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 A001077 Numerators of continued fraction convergents to sqrt(5). (Formerly M1934 N0764) 38

%I M1934 N0764

%S 1,2,9,38,161,682,2889,12238,51841,219602,930249,3940598,16692641,

%T 70711162,299537289,1268860318,5374978561,22768774562,96450076809,

%U 408569081798,1730726404001,7331474697802,31056625195209

%N Numerators of continued fraction convergents to sqrt(5).

%C a(2*n+1) with b(2*n+1) := A001076(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = -1.

%C a(2*n) with b(2*n) := A001076(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = +1 (see Emerson reference).

%C Bisection: a(2*n) = T(n,9) = A023039(n), n >= 0 and a(2*n+1) = 2*S(2*n, 2*sqrt(5)), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. See A053120, resp. A049310.

%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. 282.

%H G. C. Greubel, <a href="/A001077/b001077.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)

%H E. I. Emerson, <a href="http://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, Ex.1, p. 237-8.

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

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%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 (4,1).

%F G.f.: (1-2*x)/(1-4*x-x^2).

%F a(n) = 4*a(n-1) + a(n-2), a(0)=1, a(1)=2.

%F a(n) = ((2 + sqrt(5))^n + (2 - sqrt(5))^n)/2.

%F a(n) = A014448(n)/2.

%F Lim_{n->inf} a(n)/a(n-1) = phi^3 = 2 + sqrt(5). - _Gregory V. Richardson_, Oct 13 2002

%F a(n) = ((-i)^n)*T(n, 2*i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.

%F Binomial transform of A084057. - _Paul Barry_, May 10 2003

%F E.g.f.: exp(2x)cosh(sqrt(5)x). - _Paul Barry_, May 10 2003

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*5^k*2^(n-2k). - _Paul Barry_, Nov 15 2003

%F a(n) = 4*a(n-1) + a(n-2) when n > 2; a(1) = 1, a(2) = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 25 2004

%F a(n) = A001076(n+1) - 2*A001076(n) = A097924(n) - A015448(n+1); a(n+1) = A097924(n) + 2*A001076(n) = A097924(n) + 2(A048876(n) - A048875(n)). - _Creighton Dement_, Mar 19 2005

%F a(n) = F(3*n)/2 + F(3*n-1) where F() = Fibonacci numbers A000045. - _Gerald McGarvey_, Apr 28 2007

%F a(n) = A000032(3*n)/2.

%F For n >= 1: a(n) = 1/2*Fibonacci(6*n)/Fibonacci(3*n) and a(n) = integer part of (2 + sqrt(5))^n. - _Artur Jasinski_, Nov 28 2011

%F a(n) = Sum_{k=0..n} A201730(n,k)*4^k. - _Philippe Deléham_, Dec 06 2011

%F a(n) = A001076(n) + A015448(n). - _R. J. Mathar_, Jul 06 2012

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013

%F a(n) is the (1,1)-entry of the matrix W^n with W=[2, sqrt(5); sqrt(5), 2]. - _Carmine Suriano_, Mar 21 2014

%e 1 2 9 38 161 (A001077)

%e -,-,-,--,---, ...

%e 0 1 4 17 72 (A001076)

%e 1 + 2*x + 9*x^2 + 38*x^3 + 161*x^4 + 682*x^5 + 2889*x^6 + 12238*x^7 + ... - _Michael Somos_, Aug 11 2009

%p A001077:=(-1+2*z)/(-1+4*z+z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p with(combinat): a:=n->fibonacci(n+1, 4)-2*fibonacci(n, 4): seq(a(n), n=0..30); # _Zerinvary Lajos_, Apr 04 2008

%t LinearRecurrence[{4, 1}, {1, 2}, 30]

%t Join[{1},Numerator[Convergents[Sqrt[5],30]]](* _Harvey P. Dale_, Mar 23 2016 *)

%t CoefficientList[Series[(1-2*x)/(1-4*x-x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 19 2017

%o (Sage) [lucas_number2(n,4,-1)/2 for n in xrange(0, 23)] # _Zerinvary Lajos_, May 14 2009

%o (PARI) {a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)} /* _Michael Somos_, Aug 11 2009 */

%o (PARI) a(n)=if(n<2,n+1,my(t=4);for(i=1,n-2,t=4+1/t);numerator(2+1/t)) \\ _Charles R Greathouse IV_, Dec 05 2011

%o (PARI) x='x+O('x^30); Vec((1-2*x)/(1-4*x-x^2)) \\ _G. C. Greubel_, Dec 19 2017

%o (MAGMA) I:=[1, 2]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; (* _G. C. Greubel_, Dec 19 2017 *)

%Y Cf. A001076, A023039, A049629, A000032 (Lucas Numbers).

%K nonn,easy,frac,nice

%O 0,2

%A _N. J. A. Sloane_

%E Chebyshev comments from _Wolfdieter Lang_, Jan 10 2003

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Last modified October 19 13:38 EDT 2018. Contains 316361 sequences. (Running on oeis4.)