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 A024036 a(n) = 4^n - 1. 40
 0, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663, 281474976710655 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 iterations 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 increases. The ratio between the number of segments of the curve at n-th iteration (A015521) and a(n) tend to 4/5 as n increases. - Giorgio Balzarotti, Mar 16 2006 Numbers whose base 4 representation is 333....3. - Zerinvary Lajos, Feb 03 2007 From Eric Desbiaux, Jun 28 2009: (Start) It appears that for a given area, a square n^2 can be divided 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. - Reinhard Zumkeller, Nov 22 2009 A000120(a(n)) = 2*n. - Reinhard Zumkeller, Feb 07 2011 Also total number of line segments after n-th stage in the H tree, if 4^(n-1) H's are added at n-th stage to the structure in which every "H" is formed by 3 line segments. A164346 (the first differences of this sequence) give the number of line segments added at n-th stage. - Omar E. Pol, Feb 16 2013 a(n) is the cumulative number of segment deletions in a Koch snowflake after (n+1) iterations. - Ivan N. Ianakiev, Nov 22 2013 Inverse binomial transform of A005057. - Wesley Ivan Hurt, Apr 04 2014 For n>0, a(n) is one-third the partial sums of A002063(n-1). - J. M. Bergot, May 23 2014 Also the cyclomatic number of the n-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 18 2017 REFERENCES G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255. LINKS Felix Fröhlich, Table of n, a(n) for n = 0..99 A. V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, arXiv:1809.00122 [math.CA], 2018. Eric Weisstein's World of Mathematics, Cyclomatic Number Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph Index entries for linear recurrences with constant coefficients, signature (5,-4). FORMULA a(n) = 3 * A002450(n). - N. J. A. Sloane, Feb 19 2004 G.f.: 3*x/(-1+x)/(-1+4*x) = 1/(-1+x)-1/(-1+4*x). - R. J. Mathar, Nov 23 2007 E.g.f.: e^(4*x)-e^x. - Mohammad K. Azarian, Jan 14 2009 a(n) = A000051(n) * A000225(n). - Reinhard Zumkeller, Feb 14 2009 a(n) = A179857(A000225(n)), for n>0; a(n) > A179857(m), for m < A000225(n). - Reinhard Zumkeller, Jul 31 2010 a(n) = 4*a(n-1)+3, with a(0)=0. - Vincenzo Librandi, Aug 01 2010 a(n) = (3/2) * A020988(n). - Omar E. Pol, Mar 15 2012 a(n) = (Sum_{i=0..n} A002001(i)) - 1 = A178789(n+1) - 3. - Ivan N. Ianakiev, Nov 22 2013 a(n) = n*E(2*n-1,1)/B(2*n,1), for n>0, where E(n,x) denotes the Euler polynomials and B(n,x) the Bernoulli polynomials. - Peter Luschny, Apr 04 2014 EXAMPLE G.f. = 3*x + 15*x^2 + 63*x^3 + 255*x^4 + 1023*x^5 + 4095*x^6 + ... MAPLE A024036:=n->4^n-1; seq(A024036(n), n=0..30); # Wesley Ivan Hurt, Apr 04 2014 MATHEMATICA Array[4^# - 1 &, 50, 0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *) (* Start from Eric W. Weisstein, Sep 19 2017 *) Table[4^n - 1, {n, 0, 20}] 4^Range[0, 20] - 1 LinearRecurrence[{5, -4}, {0, 3}, 20] CoefficientList[Series[3 x/(1 - 5 x + 4 x^2), {x, 0, 20}], x] (* End *) PROG (Sage) [gaussian_binomial(2*n, 1, 2) for n in xrange(0, 21)] # Zerinvary Lajos, May 28 2009 (Sage) [stirling_number2(2*n+1, 2) for n in xrange(0, 21)] # Zerinvary Lajos, Nov 26 2009 (Haskell) a024036 = (subtract 1) . a000302 a024036_list = iterate ((+ 3) . (* 4)) 0 -- Reinhard Zumkeller, Oct 03 2012 (PARI) for(n=0, 100, print1(4^n-1, ", ")) \\ Felix Fröhlich, Jul 04 2014 CROSSREFS Cf. A015521. Sequence in context: A218236 A218282 A103454 * A111303 A118339 A083858 Adjacent sequences:  A024033 A024034 A024035 * A024037 A024038 A024039 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms Wesley Ivan Hurt, Apr 04 2014 STATUS approved

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Last modified May 26 07:44 EDT 2019. Contains 323579 sequences. (Running on oeis4.)