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

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

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, 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

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)

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

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

Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Links

Felix Fröhlich, Table of n, a(n) for n = 0..99

Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 15 (2019), 046, 53 pages; arXiv preprint, arXiv:1809.00122 [math.CA], 2018-2019.

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.: exp(4*x) - exp(x). - Mohammad K. Azarian, Jan 14 2009

a(n) = A000051(n)*A000225(n). - Reinhard Zumkeller, Feb 14 2009

A079978(a(n)) = 1. - Reinhard Zumkeller, Nov 22 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

A000120(a(n)) = 2*n. - Reinhard Zumkeller, Feb 07 2011

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

a(n) = A000302(n) - 1. - Sean A. Irvine, Jun 18 2019

Sum_{n>=1} 1/a(n) = A248721. - Amiram Eldar, Nov 13 2020

a(n) = A080674(n) - A002450(n). - Elmo R. Oliveira, Dec 02 2023

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 range(21)] # Zerinvary Lajos, May 28 2009
(Sage) [stirling_number2(2*n+1, 2) for n in range(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. 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.

Sequence in context: A218236 A218282 A103454 * A111303 A118339 A083858

Adjacent sequences: A024033 A024034 A024035 * A024037 A024038 A024039

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms Wesley Ivan Hurt, Apr 04 2014

Status

approved