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A005668
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Denominators of continued fraction convergents to sqrt(10).
(Formerly M4227)
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18
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0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, 110940200785, 683644320912, 4212806126257, 25960481078454, 159975692596981, 985814636660340
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(2*n+1) with b(2*n+1) := A005667(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 10*a^2 = -1, a(2*n) with b(2*n) := A005667(2*n), n>=1, give all (positive integer) solutions to Pell equation b^2 - 10*a^2 = +1 (cf. Emerson reference).
Bisection: a(2*n)= 6*S(n-1,2*19) = 6*A078987(n-1), n>=0 and a(2*n+1)=T(2*n+1,sqrt(10))/sqrt(10), n>=0, with S(n,x), resp. T(n,x), Chebyshev's polynomials of the second, resp. first kind. S(-1,x)=0. See A049310, resp. A053120.
Sqrt(10) = 6/2 + 6/37 + 6/(37*1405) + 6/(1405*53353)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2007
a(p) == 40^((p-1)/2)) mod p, for odd primes p. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 22 2009]
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 6'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
| 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, 33 (2007)
S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 427
Tanya Khovanova, Recursive Sequences
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: x / (1 - 6*x - x^2). a(n) = 6a(n-1)+a(n-2).
a(n) = ((-i)^(n-1))*S(n-1, 6*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.
a(n)=F(n, 6), the n-th Fibonacci polynomial evaluated at x=6. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
a(n) = ((3+Sqrt[10])^n-(3-Sqrt[10])^n)/(2Sqrt[10]); a(n) = Sum_0^{Floor[(n-1)/2]} Binomial[n-1-i,i]*6^{n-1-2i} - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Sep 24 2007
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MAPLE
| evalf(sqrt(10), 200); convert(%, confrac, fractionlist); fractionlist;
A005668:=-z/(-1+6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*6, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
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PROG
| sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(0, 1, 6, 6, 1, 0) sage: [it.next() for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
(Other) sage: [lucas_number1(n, 6, -1) for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
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CROSSREFS
| Cf. A005667, A000045, A000129, A006190, A001076, A052918.
Cf. A000045, A000129, A006190, A001076, A052918.
Sequence in context: A033124 A180032 A022035 * A018904 A192807 A076026
Adjacent sequences: A005665 A005666 A005667 * A005669 A005670 A005671
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KEYWORD
| nonn,cofr,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, R. K. Guy
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EXTENSIONS
| Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 21 2003
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