1,1

Lim. n-> Inf. a(n)/a(n-1) = 3 + 2*Sqrt(2).

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

The upper intermediate convergents to 2^(1/2) beginning with 10/7, 58/41, 338/239, 1970/1393 form a strictly decreasing sequence; essentially, numerators=A075870, denominators=A002315. - Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008

A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.

Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

J. J. O'Connor and E. F. Robertson, Pell's Equation

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a(n) = 1/sqrt(2)*((1+sqrt(2))^(2*n-1)-(1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2)

G.f.: 2x(1-x)/(1-6x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]

Cf. 2*A001653.

Sequence in context: A000172 A097971 A093303 this_sequence A074608 A086871 A108450

Adjacent sequences: A075867 A075868 A075869 this_sequence A075871 A075872 A075873

nonn

Gregory V. Richardson (omomom(AT)hotmail.com), Oct 16 2002