%I A079496
%S A079496 1,3,5,17,29,99,169,577,985,3363,5741,19601,33461,114243,195025,665857,
%T A079496 1136689,3880899,6625109,22619537,38613965,131836323,225058681,
%U A079496 768398401,1311738121,4478554083,7645370045,26102926097,44560482149
%N A079496 a(1) = 1; a(2n+1)=2a(2n)-a(2n-1), a(2n)=4a(2n-1)-a(2n-2).
%C A079496 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) 
               ).
%C A079496 a(n)*a(n+3) - a(n+1)*a(n+2) = 2. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Feb 22 2003
%C A079496 n such that floor(sqrt(2)*n^2)=n*floor(sqrt(2)*n).
%C A079496 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
%C A079496 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
%D A079496 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik, 52 (1997) 122-126.
%D A079496 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, 
               New York, 1966.
%H A079496 Yujun Yang, Heping Zhang, <a href="http://dx.doi.org/10.1002/qua.21537">
               Kirchhoff Index of linear hexagonal chains</a>, Int. J. Quant. Chem. 
               108 (2008) 503-512, eq (3.3).
%F A079496 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)
%F A079496 G.f.: (1+3x-x^2-x^3)/(1-6x^2+x^4).
%F A079496 Equals A133080 * A000129, where A000129 = the Pell numbers. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Sep 18 2007
%F A079496 a(n)=6a(n-2)-a(n-4). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 
               04 2008
%Y A079496 Cf. A133080.
%Y A079496 Cf. A058580.
%Y A079496 Sequence in context: A074931 A023226 A113169 this_sequence A038898 A089133 
               A103149
%Y A079496 Adjacent sequences: A079493 A079494 A079495 this_sequence A079497 A079498 
               A079499
%K A079496 nonn
%O A079496 1,2
%A A079496 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 20 2003