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%I #113 Dec 04 2023 12:33:03
%S 0,3,15,63,255,1023,4095,16383,65535,262143,1048575,4194303,16777215,
%T 67108863,268435455,1073741823,4294967295,17179869183,68719476735,
%U 274877906943,1099511627775,4398046511103,17592186044415,70368744177663,281474976710655
%N a(n) = 4^n - 1.
%C 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, the area of the square where the curve is generated, as n increases. The ratio between the number of segments of the curve at the n-th iteration (A015521) and a(n) tends to 4/5 as n increases. - _Giorgio Balzarotti_, Mar 16 2006
%C Numbers whose base-4 representation is 333....3. - _Zerinvary Lajos_, Feb 03 2007
%C From _Eric Desbiaux_, Jun 28 2009: (Start)
%C It appears that for a given area, a square n^2 can be divided into n^2+1 other squares.
%C It's a rotation and zoom out of a Cartesian plan, which creates squares with side
%C = sqrt( (n^2) / (n^2+1) ) --> A010503|A010532|A010541... --> limit 1,
%C and diagonal sqrt(2*sqrt((n^2)/(n^2+1))) --> A010767|... --> limit A002193.
%C (End)
%C Also the total number of line segments after the n-th stage in the H tree, if 4^(n-1) H's are added at the n-th stage to the structure in which every "H" is formed by 3 line segments. A164346 (the first differences of this sequence) gives the number of line segments added at the n-th stage. - _Omar E. Pol_, Feb 16 2013
%C a(n) is the cumulative number of segment deletions in a Koch snowflake after (n+1) iterations. - _Ivan N. Ianakiev_, Nov 22 2013
%C Inverse binomial transform of A005057. - _Wesley Ivan Hurt_, Apr 04 2014
%C For n > 0, a(n) is one-third the partial sums of A002063(n-1). - _J. M. Bergot_, May 23 2014
%C Also the cyclomatic number of the n-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Sep 18 2017
%D Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%H Felix Fröhlich, <a href="/A024036/b024036.txt">Table of n, a(n) for n = 0..99</a>
%H Alexander V. Kitaev, <a href="https://doi.org/10.3842/SIGMA.2019.046">Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin</a>, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 15 (2019), 046, 53 pages; <a href="https://arxiv.org/abs/1809.00122">arXiv preprint</a>, arXiv:1809.00122 [math.CA], 2018-2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclomaticNumber.html">Cyclomatic Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiTetrahedronGraph.html">Sierpinski Tetrahedron Graph</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F a(n) = 3*A002450(n). - _N. J. A. Sloane_, Feb 19 2004
%F G.f.: 3*x/((-1+x)*(-1+4*x)) = 1/(-1+x) - 1/(-1+4*x). - _R. J. Mathar_, Nov 23 2007
%F E.g.f.: exp(4*x) - exp(x). - _Mohammad K. Azarian_, Jan 14 2009
%F a(n) = A000051(n)*A000225(n). - _Reinhard Zumkeller_, Feb 14 2009
%F A079978(a(n)) = 1. - _Reinhard Zumkeller_, Nov 22 2009
%F a(n) = A179857(A000225(n)), for n > 0; a(n) > A179857(m), for m < A000225(n). - _Reinhard Zumkeller_, Jul 31 2010
%F a(n) = 4*a(n-1) + 3, with a(0) = 0. - _Vincenzo Librandi_, Aug 01 2010
%F A000120(a(n)) = 2*n. - _Reinhard Zumkeller_, Feb 07 2011
%F a(n) = (3/2)*A020988(n). - _Omar E. Pol_, Mar 15 2012
%F a(n) = (Sum_{i=0..n} A002001(i)) - 1 = A178789(n+1) - 3. - _Ivan N. Ianakiev_, Nov 22 2013
%F 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
%F a(n) = A000302(n) - 1. - _Sean A. Irvine_, Jun 18 2019
%F Sum_{n>=1} 1/a(n) = A248721. - _Amiram Eldar_, Nov 13 2020
%F a(n) = A080674(n) - A002450(n). - _Elmo R. Oliveira_, Dec 02 2023
%e G.f. = 3*x + 15*x^2 + 63*x^3 + 255*x^4 + 1023*x^5 + 4095*x^6 + ...
%p A024036:=n->4^n-1; seq(A024036(n), n=0..30); # _Wesley Ivan Hurt_, Apr 04 2014
%t Array[4^# - 1 &, 50, 0] (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t (* Start from _Eric W. Weisstein_, Sep 19 2017 *)
%t Table[4^n - 1, {n, 0, 20}]
%t 4^Range[0, 20] - 1
%t LinearRecurrence[{5, -4}, {0, 3}, 20]
%t CoefficientList[Series[3 x/(1 - 5 x + 4 x^2), {x, 0, 20}], x]
%t (* End *)
%o (Sage) [gaussian_binomial(2*n,1, 2) for n in range(21)] # _Zerinvary Lajos_, May 28 2009
%o (Sage) [stirling_number2(2*n+1, 2) for n in range(21)] # _Zerinvary Lajos_, Nov 26 2009
%o (Haskell)
%o a024036 = (subtract 1) . a000302
%o a024036_list = iterate ((+ 3) . (* 4)) 0
%o -- _Reinhard Zumkeller_, Oct 03 2012
%o (PARI) for(n=0, 100, print1(4^n-1, ", ")) \\ _Felix Fröhlich_, Jul 04 2014
%Y Cf. A000051, A000120, A000225, A000302, A002001, A002063, A002193, A002450, A005057, A010503, A010532, A010541, A010767, A015521, A020988, A027637 (partial products), A078904 (partial sums), A079978, A080674, A164346 (first differences), A178789, A179857, A248721.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms _Wesley Ivan Hurt_, Apr 04 2014
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