0,2

Chebyshev's polynomials T(n,x) evaluated at x=10.

The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 99*b(n)^2 = +1 with b(n)=A075843(n), n>=0.

a(11+22k)-1 and a(11+22k)+1 are consecutive odd powerful numbers. The first pair is 99612037019890+-1. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006

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. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.

"Questions D'Arithmetique", Question 3686, Solution by H.L. Mennessier, Mathesis, 65(4, Supplement) 1956, pp. 1-12.

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.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

For all members x of the sequence, 11*x^2 - 11 is a square. Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*Sqrt(11) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

a(n) = T(n, 10) = (S(n, 20)-S(n-2, 20))/2, with S(n, x) := U(n, x/2), and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n-1, 20)= A075843(n).

G.f. (1-10*x)/(1-20*x+x^2).

a(n) = (((10+3*sqrt(11))^n + (10-3*sqrt(11))^n))/2.

a(n) = sqrt(99*A075843(n)^2 + 1), (cf. Richardson comment).

A001085:=-(-1+10*z)/(1-20*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

sage: [lucas_number2(n, 20, 1)/2 for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

Cf. A090728.

Sequence in context: A007816 A126463 A097127 this_sequence A079436 A126431 A036362

Adjacent sequences: A001082 A001083 A001084 this_sequence A001086 A001087 A001088

nonn

njas

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000

Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002