0,4

For n >= 4 we have the linear recurrence a(n) = 4*a(n-2) - a(n-4). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 04 2001

Integer solutions to the equation floor(sqrt(3)*x^2)=x*floor(sqrt(3)*x) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 18 2004

For n>2, a(n) is the smallest integer > a(n-1) such that sqrt(3)*a(n) is closer to and greater than an integer than sqrt(3)*a(n-1). i.e. a(n) is the smallest integer > a(n-1) such that frac(sqrt(3)*a(n))<frac(sqrt(3)*a(n-1)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 20 2003

The lower principal and intermediate convergents to 3^(1/2), beginning with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; essentially, numerators=A143643 and denominators=A005246. - Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008

T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..500

Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, A Note on Rational Succession Rules, J. Integer Seqs., Vol. 6, 2003.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

lim n ->infinity a(2n+1)/a(2n) = (3+sqrt(3))/3 =1.5773502... lim n ->infinity a(2n)/a(2n-1) = (3+sqrt(3))/2 = 2.3660254.... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 07 2002

A101265(n) = a(n)*a(n+1). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 24 2006

a(2-n)=a(n). - Michael Somos Nov 15 2006

For n>2: a(n) = a(n-1) + SUM(a(2*k): 1 <= k < n/2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 16 2007

A005246:=-(-1-z+2*z**2+z**3)/(1-4*z**2+z**4); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for one of the leading 1's.]

(PARI) {a(n)=if(n<0, n=2-n); polcoeff((1+x-3*x^2-2*x^3)/(1-4*x^2+x^4)+x*O(x^n), n)} /* Michael Somos Nov 15 2006 */

Bisections are A001835 and A001075.

Cf. A101265.

Sequence in context: A007481 A121268 A101173 this_sequence A116406 A112843 A036651

Adjacent sequences: A005243 A005244 A005245 this_sequence A005247 A005248 A005249

easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Michael Somos, Aug 01 2001