0,2

Lim. n -> Inf. a(n)/a(n-2) = 3 + 2*Sqrt(2) = R1*R2. Lim. k -> Inf. a(2*k-1)/a(2*k) = (9 + 4*Sqrt(2))/7 = R1 (ratio #1). Lim. k -> Inf. a(2*k)/a(2*k-1) = (11 + 6*Sqrt(2))/7 = R2 (ratio #2).

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

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.

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.

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

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

J. P. Robertson, Solving the Generalized Pell Equation

For n>0, a(2n)=A046090(n)+A001653(n)+A001652(n-1); a(2n+1)=A001652(n+1)-A001652(n-1)-A001653(n-1); e.g. 53=21+29+3; 111=119-3-5 - Charlie Marion (charliem(AT)bestweb.net), Aug 14 2003

The same recurrences hold for the odd and even indices respectively : a(n+2)=6*a(n+1)-a(n), a(n+1)=3*a(n)+2*(2*a(n)^2+7)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 2007

G.f.: (x+1)^3/(x^2+2*x-1)/(x^2-2*x-1). a(n)= [ -A077985(n)-3*A077985(n-1)+3*A000129(n+1)+A000129(n)]/2. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

For x an element of A077443 and y the corresponding element of this sequence, the generalized Pell equation x^2 - 2*y^2 = 7 is satisfied.

Cf. A077443.

Sequence in context: A146901 A147477 A146677 this_sequence A147455 A146429 A018316

Adjacent sequences: A077439 A077440 A077441 this_sequence A077443 A077444 A077445

nonn

Gregory V. Richardson (omomom(AT)hotmail.com), Nov 06 2002