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A001076 Denominators of continued fraction convergents to sqrt(5).
(Formerly M3538 N1434)
85
0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148, 567451585, 2403763488, 10182505537, 43133785636, 182717648081, 774004377960, 3278735159921, 13888945017644, 58834515230497, 249227005939632, 1055742538989025 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(2*n+1) with b(2*n+1) := A001077(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = -1, a(2*n) with b(2*n) := A001077(2*n), n>=1, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = +1 (cf. Emerson reference).

Bisection: a(2*n+1)= T(2*n+1,sqrt(5))/sqrt(5)= A007805(n), n>=0 and a(2*n)=4*S(n-1,18),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. S(n,18)=A049660(n+1).

Apart from initial terms, this is the Pisot sequence E(4,17), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].

This is also the Horadam sequence (0,1,1,4), having the recurrence relation a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 1. a(n) / a(n-1) converges to 5^1/2 + 2 as n approaches infinity. 5^1/2 + 2 can also be written as (2 * Phi) + 1 and Phi^2 + Phi. - Ross La Haye, Aug 18 2003

Numerators in continued fraction [2, 4, 4, 4,...] = (1, 4, 17, 72,...) = numerators of continued fraction [4, 4, 4,...]; where the convergents to [4, 4, 4,...] = (1/4, 4/17, 17/72,...). Let X = the 2 X 2 matrix [0, 1; 1, 4]; then X^n = [a(n-1), a(n); a(n), a(n+1)]; e.g. X^3 = [4, 17; 17, 72]. Let C = the limit of a(n)/a(n-1) = 2 + sqrt(5) = 4.236067977...; then C^n = a(n+1) + (1/C)*a(n), where (1/C) = .236067977.... Example: C^3 = 76.01315556..., = 72 + 17*(.2360679....). - Gary W. Adamson, Dec 15 2007

Sqrt(5) = 4/2 + 4/17 + 4/(17*305) + 4/(305*5473) + 4/(5473*98209) +... - Gary W. Adamson, Dec 15 2007

a(p) == 20^((p-1)/2)) mod p, for odd primes p. [From Gary W. Adamson, Feb 22 2009]

A001076 == One half of even Fibonacci numbers. [From Vladimir Joseph Stephan Orlovsky, Oct 25 2009]

a(n) = A167808(3*n). [From Reinhard Zumkeller, Nov 12 2009]

For n >=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 4's along the main diagonal and 1's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 08 2011]

Moreover, a(n) is the second binomial transform of (0,1,0,5,0,25,...) (see also A033887). This fact can be proved similarly like the proof of Paul Barry's remark in A033887 by using the following scaling identity for delta-Fibonacci numbers: y^n b(n;x/y) = sum_{k=0..n} binomial(n,k) (y-1)^(n-k) b(k;x) and the fact that b(n;2) = (1-(-1)^n) 5^floor(n/2). - Roman Witula, Jul 12 2012

Binomial transform of 0, 1, 2, 8, 24, 80, 256,... (A063727 with offset 1). - R. J. Mathar, Feb 05 2014

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 23.

D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 398

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 entries for sequences related to linear recurrences with constant coefficients, signature (4,1).

FORMULA

a(n) = 4a(n-1) + a(n-2), n>1. a(0)=0, a(1)=1. G.f.: x/(1-4*x-x^2).

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

a(n)=A014445(n)/2 = F(3n)/2.

a(n) = ((-i)^(n-1))*S(n-1, 4*i), with i^2 =-1 and S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0.

a(n)=sum{i=0..n, sum{j=0..n, Fib(i+j)*n!/(i!j!(n-i-j)!)/2}} - Paul Barry, Feb 06 2004

E.g.f.: exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - Vladeta Jovovic, Sep 01 2004

a(n) = F(1) + F(4) + F(7) + ... + F(3n-2), for n>0.

Conjecture: 2a(n+1) = a(n+2) - A001077(n+1); Sequences (a(n)), A001077 generated by floretion: 'ii' + 'jj' - 'kk' + 0.5'ik' + 0.5'ki' - e - Creighton Dement, Nov 28 2004

a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)F(j)/2}} - Paul Barry, Feb 14 2005

a(n) = A048876(n) - A048875(n) - Creighton Dement, Mar 19 2005

Let M = {{0, 1}, {1, 4}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = v[n][[1]]. - Roger L. Bagula, May 29 2005 - T. D. Noe, Jan 19 2006

a(n)=F(n, 4), the n-th Fibonacci polynomial evaluated at x=4. - T. D. Noe, Jan 19 2006

[A015448(n), a(n)] = [1,4; 1,3]^n * [1,0] - Gary W. Adamson, Mar 21 2008

a(n) = sum(Fibonacci(3*k-2), k=0..n)+1. - Gary Detlefs Dec 26 2010

a(n) = (3*(-1)^n*F(n) + 5*F(n)^3)/2, n >= 0. See the general D. Jennings formula given in a comment on triangle A111125, where also the reference is given. Here the second (k=1) row [3,1] applies. - Wolfdieter Lang, Sep 01 2012

Identities: sum_{k>=1} (-1)^(k-1)/(a(k)*a(k+1)) = (sum_{k>=1} (-1)^(k-1)/(F_k*F_(k+1)))^3 = phi^(-3), where F_n is the n-th Fibonacci numbers (A000045) and phi is golden ratio (A001622). - Vladimir Shevelev, Feb 23 2013

G.f.: Q(0)*x/(2-4*x), where Q(k) = 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2013

a(-n) = -(-1)^n * a(n). - Michael Somos, Feb 23 2014

EXAMPLE

1 2 9 38 161 (A001077)

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

0 1 4 17 72 (A001076)

G.f. = x + 4*x^2 + 17*x^3 + 72*x^4 + 305*x^5 + 1292*x^6 + 5473*x^7 + 23184*x^8 + ...

MAPLE

K:=1/(1+4*z-z^2): Kser:=series(K, z=0, 30): seq(abs(coeff(Kser, z, n)), n= -1..21); - Zerinvary Lajos, Nov 08 2007

A001076:=-1/(-1+4*z+z**2); [Conjectured by Simon Plouffe in his 1992 dissertation.]

with(combinat): a:=n->fibonacci(n, 4)-4*fibonacci(n-1, 4): seq(a(n), n=2..24); - Zerinvary Lajos, Apr 04 2008

MATHEMATICA

CoefficientList[Series[-z/(z^2 + 4 z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)

Join[{0}, Denominator[Convergents[Sqrt[5], 30]]] (* Harvey P. Dale, Dec 10 2011 *)

a[ n_] := Fibonacci[ 3 n] / 2; (* Michael Somos, Feb 23 2014 *)

a[ n_] := ((2 + Sqrt[5])^n - (2 - Sqrt[5])^n) /(2 Sqrt[5]) // Simplify; (* Michael Somos, Feb 23 2014 *)

PROG

(Mupad) numlib::fibonacci(3*n)/2 $ n = 0..30; - Zerinvary Lajos, May 09 2008

(Sage) [lucas_number1(n, 4, -1) for n in xrange(0, 23)]# [From Zerinvary Lajos, Apr 23 2009]

(Sage) [fibonacci(3*n)/2 for n in xrange(0, 23)]# [From Zerinvary Lajos, May 15 2009]

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

(PARI) {a(n) = imag( (2 + quadgen(20))^n )}; /* Michael Somos, Feb 23 2014 */

CROSSREFS

Cf. A001077, A015448, A175183 (Pisano periods).

Cf. A049660, A007805.

Partial sums of A033887. First differences of A049652. Bisection of A059973.

Third column of array A028412.

Cf. A243399.

Sequence in context: A179606 A108929 A022031 * A122451 A113442 A085732

Adjacent sequences:  A001073 A001074 A001075 * A001077 A001078 A001079

KEYWORD

nonn,easy,cofr,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

STATUS

approved

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Last modified September 17 21:42 EDT 2014. Contains 246885 sequences.