Search: id:A005668 Results 1-1 of 1 results found. %I A005668 M4227 %S A005668 0,1,6,37,228,1405,8658,53353,328776,2026009,12484830,76934989, %T A005668 474094764,2921503573,18003116202,110940200785,683644320912, %U A005668 4212806126257,25960481078454,159975692596981,985814636660340 %N A005668 Denominators of continued fraction convergents to sqrt(10). %C A005668 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). %C A005668 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. %C A005668 Sqrt(10) = 6/2 + 6/37 + 6/(37*1405) + 6/(1405*53353)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2007 %C A005668 a(p) == 40^((p-1)/2)) mod p, for odd primes p. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 22 2009] %D A005668 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A005668 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. %D A005668 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233. %D A005668 S. Falcon & A. Plaza: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33 (2007) %D A005668 S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007) %H A005668 Index entries for sequences related to linear recurrences with constant coefficients %H A005668 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. %H A005668 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A005668 Tanya Khovanova, Recursive Sequences %H A005668 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 427 %H A005668 Index entries for sequences related to Chebyshev polynomials. %F A005668 G.f.: x / (1 - 6*x - x^2). a(n) = 6a(n-1)+a(n-2). %F A005668 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. %F A005668 a(n)=F(n, 6), the n-th Fibonacci polynomial evaluated at x=6. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006 %F A005668 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 %p A005668 evalf(sqrt(10),200); convert(%,confrac,fractionlist); fractionlist; %p A005668 A005668:=-z/(-1+6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.] %t A005668 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] %o A005668 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 %o A005668 (Other) sage: [lucas_number1(n,6,-1) for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009] %Y A005668 Cf. A045667, A000045, A000129, A006190, A001076, A052918. %Y A005668 Cf. A000045, A000129, A006190, A001076, A052918. %Y A005668 Sequence in context: A033116 A033124 A022035 this_sequence A018904 A076026 A161734 %Y A005668 Adjacent sequences: A005665 A005666 A005667 this_sequence A005669 A005670 A005671 %K A005668 nonn,cofr,easy %O A005668 0,3 %A A005668 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, R. K. Guy %E A005668 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 21 2003 Search completed in 0.002 seconds