1,2

a(1)=1, a(n) is the smallest integer > a(n-1) such that sqrt(2)*a(n) is closer and > to an integer than sqrt(2)*a(n-1) ( i.e. a(n) is the smallest integer > a(n-1) such that frac(sqrt(2)*a(n))<frac(sqrt(2)*a(n-1) ).

a(n)*a(n+3) - a(n+1)*a(n+2) = 2. - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 22 2003

n such that floor(sqrt(2)*n^2)=n*floor(sqrt(2)*n).

The sequence 1,1,3,5,17.... has g.f. (1+x-3x^2-x^3)/(1-6x^2+x^4); a(n)=sum{k=0..floor(n/2), C(n,2k)2^(n-k-floor((n+1)/2))}; a(n)=-(sqrt(2)-1)^n((sqrt(2)/8-1/4)(-1)^n-sqrt(2)/8-1/4)-(sqrt(2)+1)^n((sqrt(2)/8-1/4)(-1)^n-sqrt(2)/8-1/4); a(2n)=A001541(n)=A001333(2n); a(2n+1)=A001653(n)=A000129(2n+1). - Paul Barry (pbarry(AT)wit.ie), Jan 22 2005

The lower principal and intermediate convergents to 2^(1/2), beginning with 1/1, 4/3, 7/5, 24/17, 41/29, form a strictly increasing sequence; essentially, numerators=A143608 and denominators=A079496. - Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008

Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Yujun Yang, Heping Zhang, Kirchhoff Index of linear hexagonal chains, Int. J. Quant. Chem. 108 (2008) 503-512, eq (3.3).

a(2n+1)-a(2n)=a(2n)-a(2n-1)=A001542(n); a(2n+1)=ceiling((2+sqrt(2))/4*(3+2*sqrt(2))^n) and a(2n)=ceiling(1/2*(3+2*sqrt(2))^n)

G.f.: (1+3x-x^2-x^3)/(1-6x^2+x^4).

Equals A133080 * A000129, where A000129 = the Pell numbers. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 18 2007

a(n)=6a(n-2)-a(n-4). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 04 2008

Cf. A133080.

Cf. A058580.

Sequence in context: A074931 A023226 A113169 this_sequence A038898 A089133 A103149

Adjacent sequences: A079493 A079494 A079495 this_sequence A079497 A079498 A079499

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 20 2003