|
| |
|
|
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).
|
|
11
|
|
|
|
1, 2, 7, 12, 41, 70, 239, 408, 1393, 2378, 8119, 13860, 47321, 80782, 275807, 470832, 1607521, 2744210, 9369319, 15994428, 54608393, 93222358, 318281039, 543339720, 1855077841, 3166815962, 10812186007, 18457556052, 63018038201, 107578520350
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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, Sep 18 2007
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, Jan 28 2008
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, Aug 27 2008
|
|
|
REFERENCES
|
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.
|
|
|
LINKS
|
Table of n, a(n) for n=1..30.
Index to sequences with linear recurrences with constant coefficients, signature (0,6,0,-1).
|
|
|
FORMULA
|
"A Diofloortin equation" : n such that 2*n^2=floor(n*sqrt(2)*ceil(n*sqrt(2))).
a(n)a(n+3) = -2 + a(n+1)a(n+2).
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, Jun 06 2006
Equals A133566 * A000129, where A000129 = the Pell sequence. - Gary W. Adamson, Sep 18 2007
|
|
|
CROSSREFS
|
Cf. A084069, A084070.
Bisections are A001542 and A002315.
Cf. A133566, A079496.
Sequence in context: A073710 A092831 A055257 * A192772 A046243 A103886
Adjacent sequences: A084065 A084066 A084067 * A084069 A084070 A084071
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Benoit Cloitre, May 10 2003
|
|
|
STATUS
|
approved
|
| |
|
|
