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A079496
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a(1) = 1; a(2n+1)=2a(2n)-a(2n-1), a(2n)=4a(2n-1)-a(2n-2).
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8
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1, 3, 5, 17, 29, 99, 169, 577, 985, 3363, 5741, 19601, 33461, 114243, 195025, 665857, 1136689, 3880899, 6625109, 22619537, 38613965, 131836323, 225058681, 768398401, 1311738121, 4478554083, 7645370045, 26102926097, 44560482149
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
Contribution from Richard Choulet (richardchoulet(AT)yahoo.fr), May 09 2010: (Start)
This sequence is a particular case of the following situation: a(0)=1, a(1)=a, a(2)=b
with the recurrence relation a(n+3)=(a(n+2)*a(n+1)+q)/a(n) where q is given in Z to have Q=(a*b^2+q*b+a+q)/(a*b) itself in Z.
The g.f is f: f(z)=(1+a*z+(b-Q)*z^2+(a*b+q-a*Q)*z^3)/(1-Q*z^2+z^4); so we have the linear recurrence: a(n+4)=Q*a(n+2)-a(n).
The general form of a(n) is given by:
a(2*m)=sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(b-Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)) and
a(2*m+1)=a*sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(a*b+q-a*Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2))
[From Richard Choulet (richardchoulet(AT)yahoo.fr), Feb 24 2010] (End)
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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.
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LINKS
| John M. Campbell, An Integral Representation of Kekule' Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399, 2011.
Yujun Yang, Heping Zhang, Kirchhoff Index of linear hexagonal chains, Int. J. Quant. Chem. 108 (2008) 503-512, eq (3.3).
Index to sequences with linear recurrences with constant coefficients, signature (0,6,0,-1).
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FORMULA
| 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
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CROSSREFS
| Cf. A133080.
Cf. A058580.
Sequence in context: A023226 A113169 A174913 * A038898 A089133 A103149
Adjacent sequences: A079493 A079494 A079495 * A079497 A079498 A079499
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 20 2003
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