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A001077 Numerators of continued fraction convergents to sqrt(5).
(Formerly M1934 N0764)
36
1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, 3940598, 16692641, 70711162, 299537289, 1268860318, 5374978561, 22768774562, 96450076809, 408569081798, 1730726404001, 7331474697802, 31056625195209 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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).

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.

REFERENCES

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. 282.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Ex.1, p. 237-8.

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Chebyshev polynomials.

Index to sequences with linear recurrences with constant coefficients, signature (4,1).

FORMULA

G.f.: (1-2*x)/(1-4*x-x^2); a(n)=4*a(n-1)+a(n-2), a(0)=1, a(1)=2; a(n)=[ (2+sqrt(5))^n + (2-sqrt(5))^n ]/2.

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

Lim. n-> Inf. a(n)/a(n-1) = phi^3 = 2 + Sqrt(5). - Gregory V. Richardson, Oct 13 2002

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

Binomial transform of A084057. - Paul Barry, May 10 2003

E.g.f.: exp(2x)cosh(sqrt(5)x). - Paul Barry, May 10 2003

a(n) = sum{k=0..floor(n/2), C(n, 2k)5^k2^(n-2k)}. - Paul Barry, Nov 15 2003

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

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

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

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

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

a(n) = Sum_{k, 0<=k<=n} A201730(n,k)*4^k. - Philippe Deléham, Dec 06 2011

a(n) = A001076(n) + A015448(n). - R. J. Mathar, Jul 06 2012

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

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

EXAMPLE

1 2 9 38 161 (A001077)

-,-,-,--,---, ...

0 1 4 17 72 (A001076)

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

MAPLE

A001077:=(-1+2*z)/(-1+4*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

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

MATHEMATICA

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

PROG

(Sage) [lucas_number2(n, 4, -1)/2 for n in xrange(0, 23)] #  Zerinvary Lajos, May 14 2009

(PARI) {a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)} /* Michael Somos, Aug 11 2009 */

(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

CROSSREFS

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

Sequence in context: A181339 A037489 A037569 * A150993 A150994 A150995

Adjacent sequences:  A001074 A001075 A001076 * A001078 A001079 A001080

KEYWORD

nonn,easy,frac,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified October 24 03:14 EDT 2014. Contains 248491 sequences.