0,2

This sequence is the normalized length per iteration of the space-filling Peano-Hilbert curve. The curve remains in a square, but its length increases without bound. The length of the curve, after n iteration in a unit square, is a(n)*2^(-n) where a(n) = 4*a(n-1)+3. This is the sequence of a(n) values. a(n)*(2^(-n)*2^(-n)) tends to 1, area of the square where the curve is generated, as n increase. The ratio between the number of segments of the curve at n-th iteration (A015521) and a(n) tend to 4/5 as n increase. - Giorgio Balzarotti (greenblue(AT)tiscali.it), Mar 16 2006

Numbers whose base 4 representation is 333....3. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 03 2007

a(n) = A000051(n)*A000225(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 14 2009]

Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Jun 28 2009: (Start)

It appears that for a same area, a square n^2 can be divide into n^2+1 other squares.

It's a rotation and zoom out of a cartesian plan,

which creates squares with side

= sqrt( (n^2) / (n^2+1) ) --> A010503|A010532|A010541... --> limit 1,

and diagonal sqrt(2*sqrt((n^2)/(n^2+1))) --> A010767|... --> limit A002193

(End)

A079978(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 22 2009]

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

G.f.: 3*x/(-1+x)/(-1+4*x) = 1/(-1+x)-1/(-1+4*x) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2007

E.g.f.: e^(4*x)-e^x. [From Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 14 2009]

a(n)=4*a(n-1)+3 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 30 2009]

For n=2, a(2)=4*0+3=3; n=3, a(3)=4*3+3=15; n=4, a(4)=4*15+3=63 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 30 2009]

Array[4^#-1&, 50, 0] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 03 2009]

(Other) sage: [gaussian_binomial(2*n, 1, 2) for n in xrange(0, 21)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]

(Other) sage: [stirling_number2(2*n+1, 2) for n in xrange(0, 21)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2009]

Equals 3 * A002450(n).

Cf. A015521.

Sequence in context: A122671 A067562 A062211 this_sequence A103454 A111303 A118339

Adjacent sequences: A024033 A024034 A024035 this_sequence A024037 A024038 A024039

nonn,new

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