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A024036 a(n) = 4^n - 1.

%I A024036 #114 Feb 16 2025 08:32:34
%S A024036 0,3,15,63,255,1023,4095,16383,65535,262143,1048575,4194303,16777215,
%T A024036 67108863,268435455,1073741823,4294967295,17179869183,68719476735,
%U A024036 274877906943,1099511627775,4398046511103,17592186044415,70368744177663,281474976710655
%N A024036 a(n) = 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 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 A024036 Numbers whose base-4 representation is 333....3. - _Zerinvary Lajos_, Feb 03 2007
%C A024036 From _Eric Desbiaux_, Jun 28 2009: (Start)
%C A024036 It appears that for a given area, a square n^2 can be divided into n^2+1 other squares.
%C A024036 It's a rotation and zoom out of a Cartesian plan, 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 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 A024036 a(n) is the cumulative number of segment deletions in a Koch snowflake after (n+1) iterations. - _Ivan N. Ianakiev_, Nov 22 2013
%C A024036 Inverse binomial transform of A005057. - _Wesley Ivan Hurt_, Apr 04 2014
%C A024036 For n > 0, a(n) is one-third the partial sums of A002063(n-1). - _J. M. Bergot_, May 23 2014
%C A024036 Also the cyclomatic number of the n-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Sep 18 2017
%D A024036 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%H A024036 Felix Fröhlich, <a href="/A024036/b024036.txt">Table of n, a(n) for n = 0..99</a>
%H A024036 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 A024036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclomaticNumber.html">Cyclomatic Number</a>.
%H A024036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiTetrahedronGraph.html">Sierpinski Tetrahedron Graph</a>.
%H A024036 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F A024036 a(n) = 3*A002450(n). - _N. J. A. Sloane_, Feb 19 2004
%F A024036 G.f.: 3*x/((-1+x)*(-1+4*x)) = 1/(-1+x) - 1/(-1+4*x). - _R. J. Mathar_, Nov 23 2007
%F A024036 E.g.f.: exp(4*x) - exp(x). - _Mohammad K. Azarian_, Jan 14 2009
%F A024036 a(n) = A000051(n)*A000225(n). - _Reinhard Zumkeller_, Feb 14 2009
%F A024036 A079978(a(n)) = 1. - _Reinhard Zumkeller_, Nov 22 2009
%F A024036 a(n) = A179857(A000225(n)), for n > 0; a(n) > A179857(m), for m < A000225(n). - _Reinhard Zumkeller_, Jul 31 2010
%F A024036 a(n) = 4*a(n-1) + 3, with a(0) = 0. - _Vincenzo Librandi_, Aug 01 2010
%F A024036 A000120(a(n)) = 2*n. - _Reinhard Zumkeller_, Feb 07 2011
%F A024036 a(n) = (3/2)*A020988(n). - _Omar E. Pol_, Mar 15 2012
%F A024036 a(n) = (Sum_{i=0..n} A002001(i)) - 1 = A178789(n+1) - 3. - _Ivan N. Ianakiev_, Nov 22 2013
%F A024036 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 A024036 a(n) = A000302(n) - 1. - _Sean A. Irvine_, Jun 18 2019
%F A024036 Sum_{n>=1} 1/a(n) = A248721. - _Amiram Eldar_, Nov 13 2020
%F A024036 a(n) = A080674(n) - A002450(n). - _Elmo R. Oliveira_, Dec 02 2023
%e A024036 G.f. = 3*x + 15*x^2 + 63*x^3 + 255*x^4 + 1023*x^5 + 4095*x^6 + ...
%p A024036 A024036:=n->4^n-1; seq(A024036(n), n=0..30); # _Wesley Ivan Hurt_, Apr 04 2014
%t A024036 Array[4^# - 1 &, 50, 0] (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t A024036 (* Start from _Eric W. Weisstein_, Sep 19 2017 *)
%t A024036 Table[4^n - 1, {n, 0, 20}]
%t A024036 4^Range[0, 20] - 1
%t A024036 LinearRecurrence[{5, -4}, {0, 3}, 20]
%t A024036 CoefficientList[Series[3 x/(1 - 5 x + 4 x^2), {x, 0, 20}], x]
%t A024036 (* End *)
%o A024036 (Sage) [gaussian_binomial(2*n,1, 2) for n in range(21)] # _Zerinvary Lajos_, May 28 2009
%o A024036 (Sage) [stirling_number2(2*n+1, 2) for n in range(21)] # _Zerinvary Lajos_, Nov 26 2009
%o A024036 (Haskell)
%o A024036 a024036 = (subtract 1) . a000302
%o A024036 a024036_list = iterate ((+ 3) . (* 4)) 0
%o A024036 -- _Reinhard Zumkeller_, Oct 03 2012
%o A024036 (PARI) for(n=0, 100, print1(4^n-1, ", ")) \\ _Felix Fröhlich_, Jul 04 2014
%Y A024036 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 A024036 nonn,easy
%O A024036 0,2
%A A024036 _N. J. A. Sloane_
%E A024036 More terms _Wesley Ivan Hurt_, Apr 04 2014