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A041025
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Denominators of continued fraction convergents to sqrt(17).
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12
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1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, 151693352, 1232221121, 10009462320, 81307919681, 660472819768, 5365090477825, 43581196642368, 354014663616769, 2875698505576520, 23359602708228929, 189752520171407952, 1541379764079492545
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(2*n+1) with b(2*n+1) := A041024(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 17*a^2 = +1, a(2*n) with b(2*n) := A041024(2*n), n>=0, give all (positive integer) solutions to Pell equation b^2 - 17*a^2 = -1 (cf. Emerson reference).
Bisection: a(2*n)= T(2*n+1,sqrt(17))/sqrt(17)= A078988(n), n>=0 and a(2*n+1)=8*S(n-1,66),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.
Sqrt(17) = 8/2 + 8/65 + 8/(65*4289) + 8/(4289*283009)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2007
a(p) == ((p-1)/2)) mod p for odd primes p. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 22 2009]
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 8's along the main diagonal and 1's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 08, 2011]
De Moivres formula: a(n) = (r^n-s^n)/(r-s), for r>s, gives sequences with integer numbers if r and s are conjugates. With r=4+sqrt(17) and s=4-sqrt(17), a(n+1)/a(n) converges to r=4+sqrt(17). - Sture Sjöstedt, Nov 11 2011
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REFERENCES
| E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.
S. Falcon & A. Plaza: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals (2007)
S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007)
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: 1/(1-8*x-x^2).
a(n) = ((-i)^n)*S(n, 8*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind and i^2=-1. See A049310.
a(n)=F(n, 8), the n-th Fibonacci polynomial evaluated at x=8. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
a(n) = ((4+Sqrt[17])^n-(4-Sqrt[17])^n)/(2Sqrt{17]); a(n) = Sum[Binomial[n-1-i,i]*8^{n-1-2i}, {i,0,Floor[(n-1)/2]}] - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Sep 24 2007
Let T = the 2 X 2 matrix [0, 1; 1, 8]. Then T^n * [1, 0] = [a(n-2), a(n-1)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2007
a(n)=8*a(n-1)+a(n-2), n>1 ; a(0)=1, a(1)=8. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 20 2008]
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MATHEMATICA
| a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*8, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
CoefficientList[Series[1/(-z^2 - 8 z + 1), {z, 0, 200}], z] (* From Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[17], 30]] (* From Harvey P. Dale, Aug 15 2011 *)
LinearRecurrence[{8, 1}, {1, 8}, 50] (* Sture Sjöstedt, Nov 11 2011 *)
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PROG
| (Other) sage: [lucas_number1(n, 8, -1) for n in xrange(1, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2009]
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CROSSREFS
| Cf. A041024.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413.
Sequence in context: A033118 A033126 A022039 * A163459 A081190 A189431
Adjacent sequences: A041022 A041023 A041024 * A041026 A041027 A041028
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KEYWORD
| nonn,cofr,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
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