0,2

Chebyshev's polynomials T(n,x) evaluated at x=8.

The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 63*b(n)^2 = +1 with b(n)= A077412(n-1), n>=1 and b(0)=0.

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(7) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 14 2004

a(7+14k)-1 and a(7+14k)+1 are consecutive odd powerful numbers. The first pair is 130576328+-1. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006

Except for the first term of [A001080] and of [A001081], if X=[A001081] (1,8,127,2024,32257,..,); Y=[A001080] (0,3,48,765,1192,..,) and A=[A010727] (7,7,7,..,) we have, for all terms, Pell's equation X^2-A*Y^2=1. Example: 8^2-7*3^2=1; 127^2-7*48^2=1; 2024^2-7*765^2=1; 32257^2-7*12192^2=1; [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 16 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 281.

Index entries for sequences related to linear recurrences with constant coefficients

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.

For all members x of the sequence, 7*x^2 - 7 is a square. Lim. n-> Inf. a(n)/a(n-1) = 8 + 3*Sqrt(7). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

a(n) = T(n, 8) = (S(n, 16)-S(n-2, 16))/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, 16)= A077412(n).

a(n) = ((8+3*sqrt(7))^n + (8-3*sqrt(7))^n)/2.

a(n) = sqrt(63*A077412(n-1)^2 + 1), n>=1, (cf. Richardson comment).

G.f.: (1-8*x)/(1-16*x+x^2).

A001081:=-(-1+8*z)/(1-16*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

sage: [lucas_number2(n, 16, 1)/2 for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

Cf. A090727.

Cf. A001080, A010727 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 16 2009]

Sequence in context: A035130 A055762 A029472 this_sequence A034220 A034239 A093586

Adjacent sequences: A001078 A001079 A001080 this_sequence A001082 A001083 A001084

nonn

N. J. A. Sloane (njas(AT)research.att.com).

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