Search: id:A084068
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%I A084068
%S A084068 1,2,7,12,41,70,239,408,1393,2378,8119,13860,47321,80782,275807,470832,
%T A084068 1607521,2744210,9369319,15994428,54608393,93222358,318281039,543339720,
%U A084068 1855077841,3166815962,10812186007,18457556052,63018038201,107578520350
%N A084068 a(1) = 1, a(2) = 2, a(2n) = 2*a(2n-1)-a(2n-2); a(2n+1) = 4*a(2n)-a(2n-1).
%C A084068 A000129(n+1) = A079496(n) + a(n), where A079496 = (1, 3, 5, 17, 29, 99,
               ...). Example: A000129(5) = 29 = A079496(4) + a(4) = 17 + 12. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.ciom), Sep 18 2007
%C A084068 Apart from the first two terms (1, 2) the sequence gives the numbers 
               k which are perfect medians. Namely: if k is even -> sum_{j=2, 4, 
               6, .., k-2} {j} = sum_{j=k+2, k+4, k+6,..k+m} {j} (for some m even); 
               if k is odd -> sum_{j=1, 3, 5, .., k-2} {j} = sum_{j=k+2, k+4, k+6,
               ..k+m} {j} (for some m even). See also A001109. - Paolo P. Lava (ppl(AT)spl.at), 
               Jan 28 2008
%C A084068 The upper principal and intermediate convergents to 2^(1/2), beginning 
               with 2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence; 
               essentially, numerators=A143609 and denominators=A084068. - Clark 
               Kimberling (ck6(AT)evansville.edu), Aug 27 2008
%D A084068 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik, 52 (1997) 122-126.
%D A084068 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, 
               New York, 1966.
%F A084068 "A Diofloortin equation" : n such that 2*n^2=floor(n*sqrt(2)*ceil(n*sqrt(2))).
%F A084068 a(n)a(n+3) = -2 + a(n+1)a(n+2).
%F A084068 G.f.: x(1+x)^2/(1-6x^2+x^4); a(n)=((sqrt(2)+1)^n-(sqrt(2)-1)^n)((sqrt(2)/
               8-1/4)*(-1)^n+sqrt(2)/8+1/4); a(n+1)=sum{k=0..floor((n+1)/2), 2^k*(C(n+1,
               2k)-C(n,2k+1)*(1-(-1)^n)/2}; - Paul Barry (pbarry(AT)wit.ie), Jun 
               06 2006
%F A084068 Equals A133566 * A000129, where A000129 = the Pell sequence. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.ciom), Sep 18 2007
%Y A084068 Cf. A084069, A084070.
%Y A084068 Bisections are A001542 and A002315.
%Y A084068 Cf. A133566, A079496.
%Y A084068 Sequence in context: A073710 A092831 A055257 this_sequence A046243 A103886 
               A130710
%Y A084068 Adjacent sequences: A084065 A084066 A084067 this_sequence A084069 A084070 
               A084071
%K A084068 nonn
%O A084068 1,2
%A A084068 Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003

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