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A001076
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Denominators of continued fraction convergents to sqrt(5).
(Formerly M3538 N1434)
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79
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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
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OFFSET
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0,3
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COMMENTS
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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 halfs 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]
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REFERENCES
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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. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 282.
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LINKS
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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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,1).
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FORMULA
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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 (creighton.k.dement(AT)uni-oldenburg.de), 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 (crowdog(AT)t-online.de), 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
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EXAMPLE
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1 2 9 38 161 (A001077)
-,-,-,--,---, ...
0 1 4 17 72 (A001076)
x + 4*x^2 + 17*x^3 + 72*x^4 + 305*x^5 + 1292*x^6 + 5473*x^7 + 23184*x^8 + ... - Michael Somos Aug 11 2009
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MAPLE
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K:=1/(1+4*z-z^2): Kser:=series(K, z=0, 30): seq(abs(coeff(Kser, z, n)), n= -1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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 (zerinvarylajos(AT)yahoo.com), Apr 04 2008
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MATHEMATICA
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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 *)
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PROG
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(Mupad) numlib::fibonacci(3*n)/2 $ n = 0..30; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008
(Sage) [lucas_number1(n, 4, -1) for n in xrange(0, 23)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
(Sage) [fibonacci(3*n)/2 for n in xrange(0, 23)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
(PARI) {a(n) = fibonacci(3*n) / 2} /* Michael Somos, Aug 11 2009 */
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CROSSREFS
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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.
Sequence in context: A179606 A108929 A022031 * A122451 A113442 A085732
Adjacent sequences: A001073 A001074 A001075 * A001077 A001078 A001079
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KEYWORD
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nonn,easy,cofr,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Chebyshev comments from Wolfdieter Lang, Jan 10 2003
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STATUS
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approved
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