0,1

Positive values of x satisfying x^2 - 21*y^2 = 4; values of y are in A004254. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002

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.

J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.

T. D. Noe, Table of n, a(n) for n=0..200

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 recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

a(n) = 5*S(n-1, 5) - 2*S(n-2, 5) = S(n, 5) - S(n-2, 5) = 2*T(n, 5/2), with S(n, x)=U(n, x/2), S(-1, x)=0, S(-2, x)=-1. U(n, x), resp. T(n, x), are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 5) = A004254(n), n>=0.

G.f.: (2-5*x)/(1-5*x+x^2).

a(n) ~ (1/2*(5 + sqrt(21)))^n - Joe Keane (jgk(AT)jgk.org), May 16 2002

a(n) = ap^n + am^n, with ap=(5+sqrt(21))/2 and am=(5-sqrt(21))/2.

A003501:=-(-2+5*z)/(1-5*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

a[0] = 2; a[1] = 5; a[n_] := 5a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 21}] (from Robert G. Wilson v Jan 30 2004)

(PARI) a(n)=if(n<0, 0, subst(poltchebi(n), x, 5/2)*2)

sage: [lucas_number2(n, 5, 1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

Cf. A004252, A004253. a(n) = sqrt(4+21*A004254(n)^2).

Adjacent sequences: A003498 A003499 A003500 this_sequence A003502 A003503 A003504

Sequence in context: A106858 A100299 A038833 this_sequence A006990 A032182 A136731

nonn

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 31 2000

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002