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A079496 a(0) = a(1) = 1; thereafter a(2*n+1) = 2*a(2*n) - a(2*n-1), a(2*n) = 4*a(2*n-1) - a(2*n-2). 12
1, 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; text; internal format)
OFFSET

0,3

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, 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, 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, Aug 27 2008

From Richard Choulet, 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))

(End)

First differences of A216134, indices of the Sophie Germain triangular numbers (A124174). - Raphie Frank, Jan 04 2013

The integer square roots of floor(n^2/2 + 1) or (A007590 + 1). - Richard R. Forberg, Aug 01 2013

REFERENCES

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

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..2608

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.

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

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

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

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), a(2n) = ceiling(1/2*(3+2*sqrt(2))^n).

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

Equals A133080 * A000129. - Gary W. Adamson, Sep 18 2007

a(n) = 6*a(n-2) - a(n-4). - R. J. Mathar, Apr 04 2008

a(-n) = a(n) = A010914(n-3)*2^floor((4 - n)/2). - Michael Somos, Sep 03 2013

a(n) = (sqrt(2)*sqrt(2+(3-2*sqrt(2))^n+(3+2*sqrt(2))^n))/(2+sqrt(2)+(-1)^n*(-2+sqrt(2))). - Gerry Martens, Jun 06 2015

a(n) = 2^(n - 1)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], -1). - Peter Luschny, Sep 03 2019

EXAMPLE

1 + x + 3*x^2 + 5*x^3 + 17*x^4 + 29*x^5 + 99*x^6 + 169*x^7 + 577*x^8 + ...

MAPLE

H := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -1):

a := n -> `if`(n < 3, [1, 1, 3][n+1], 2^(n - 1)*H(n, irem(n, 2), 1/2)):

seq(simplify(a(n)), n=0..26); # Peter Luschny, Sep 03 2019

MATHEMATICA

a[1] = 1; a[2] = 3; a[3] = 5; a[n_] := a[n] = (a[n-1]*a[n-2] + 2) / a[n-3]; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Jul 17 2013, after Paul D. Hanna *)

PROG

(PARI) {a(n) = n = abs(n); 2^((4-n)\2) * real( (10 + 7 * quadgen(8)) / 2 * (2 + quadgen(8))^(n-3) ) }  /* Michael Somos, Sep 03 2013 */

(PARI) {a(n) = polcoeff( (1 + x - 3*x^2 - x^3) / (1 - 6*x^2 + x^4) + x * O(x^abs(n)), abs(n))} /* Michael Somos, Sep 03 2013 */

(MAGMA) [1, 1] cat [Floor((Sqrt(2)*Sqrt(2+(3-2*Sqrt(2))^n+(3+2*Sqrt(2))^(1+n)))/(2+Sqrt(2)-(-1)^n*(-2+Sqrt(2)))): n in [1..40]]; // Vincenzo Librandi, Jun 07 2015

CROSSREFS

Cf. A010914, A058580, A133080.

Sequence in context: A023226 A113169 A174913 * A230639 A038898 A297175

Adjacent sequences:  A079493 A079494 A079495 * A079497 A079498 A079499

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Jan 20 2003

EXTENSIONS

a(0)=1 added by Michael Somos, Sep 03 2013

STATUS

approved

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Last modified February 26 12:33 EST 2020. Contains 332279 sequences. (Running on oeis4.)