%I A024036
%S A024036 0,3,15,63,255,1023,4095,16383,65535,262143,1048575,4194303,
%T A024036 16777215,67108863,268435455,1073741823,4294967295,17179869183,
%U A024036 68719476735,274877906943,1099511627775
%N A024036 4^n-1.
%C A024036 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
%C A024036 Numbers whose base 4 representation is 333....3. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 03 2007
%C A024036 a(n) = A000051(n)*A000225(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 14 2009]
%C A024036 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Jun 28 2009: 
               (Start)
%C A024036 It appears that for a same area, a square n^2 can be divide into n^2+1 
               other squares.
%C A024036 It's a rotation and zoom out of a cartesian plan,
%C A024036 which creates squares with side
%C A024036 = sqrt( (n^2) / (n^2+1) ) --> A010503|A010532|A010541... --> limit 1,
%C A024036 and diagonal sqrt(2*sqrt((n^2)/(n^2+1))) --> A010767|... --> limit A002193
%C A024036 (End)
%C A024036 A079978(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 22 2009]
%D A024036 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%F A024036 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
%F A024036 E.g.f.: e^(4*x)-e^x. [From Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Jan 14 2009]
%F A024036 a(n)=4*a(n-1)+3 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Oct 30 2009]
%e A024036 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]
%t A024036 Array[4^#-1&,50,0] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Nov 03 2009]
%o A024036 (Other) sage: [gaussian_binomial(2*n,1,2) for n in xrange(0,21)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
%Y A024036 Equals 3 * A002450(n).
%Y A024036 Cf. A015521.
%Y A024036 Sequence in context: A122671 A067562 A062211 this_sequence A103454 A111303 
               A118339
%Y A024036 Adjacent sequences: A024033 A024034 A024035 this_sequence A024037 A024038 
               A024039
%K A024036 nonn,new
%O A024036 0,2
%A A024036 N. J. A. Sloane (njas(AT)research.att.com).