A141725 a(n) = 4^(n+1) - 3.

1, 13, 61, 253, 1021, 4093, 16381, 65533, 262141, 1048573, 4194301, 16777213, 67108861, 268435453, 1073741821, 4294967293, 17179869181, 68719476733, 274877906941, 1099511627773, 4398046511101, 17592186044413, 70368744177661

Offset

0,2

Comments

Inverse binomial transform yields A003946 with A003946(1)=4 deleted. - R. J. Mathar, Sep 13 2008

Starting with n=1, binary numbers of the form 1X01 where X is an odd number of 1's. - Brad Clardy, Mar 22 2011

Column 4 of A193871. - Omar E. Pol, Aug 22 2011

The Sierpinski square curve is a representation of this sequence, where a(n) is the number squares filled by the Sierpinski (space-filling) square curve. The square footprint expands at a rate of (2^n-1)^2 (A000225)^2. The number of nodes per iteration grows at a rate of (4^n-1)/3 (A002450). See illustration in links. - John Elias, Jul 25 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

John Elias, Illustration of Initial Terms: Sierpinski Square Curve

Mattia Fregola, Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Formula

a(n) = 10*A001045(2*n) + A001045(2*n+1).

a(n) = 4*a(n-1) + 9 for n > 0, a(0) = 1.

a(n) = A036563(2*n+2).

G.f.: (1 + 8*x)/((1 - x)*(1 - 4*x)). - R. J. Mathar, Sep 13 2008

a(n) = 4^n - 3, with offset 1. - Omar E. Pol, Aug 22 2011

a(n) = 5*a(n-1) - 4*a(n-2) for n > 1, a(0) = 1, a(1) = 13. - Harvey P. Dale, Sep 25 2011

E.g.f.: exp(4*x) - 3*exp(x). - Elmo R. Oliveira, Nov 15 2023

Maple

a:= n-> 4^(n+1)-3: seq(a(n), n=0..25); # Muniru A Asiru, Feb 20 2018

Mathematica

4^(Range[2, 25]-1)-3 (* or *) LinearRecurrence[{5, -4}, {1, 13}, 25] (* or *) NestList[4#+9&, 1, 25] (* Harvey P. Dale, Sep 25 2011 *)

Prog

(Magma) [4^(n+1)-3: n in [0..30]]; // Vincenzo Librandi, Aug 08 2011
(PARI) a(n)=4^(n+1)-3 \\ Charles R Greathouse IV, Oct 07 2015
(GAP) List([0..25], n -> 4^(n+1)-3); # Muniru A Asiru, Feb 20 2018

Crossrefs

Cf. A001045, A036563.

Sequence in context: A252970 A047673 A231800 * A279762 A147185 A122885

Adjacent sequences: A141722 A141723 A141724 * A141726 A141727 A141728

Keyword

nonn,easy

Author

Paul Curtz, Sep 13 2008

Extensions

Edited by N. J. A. Sloane, Sep 13 2008

More terms from R. J. Mathar, Sep 13 2008

Status

approved