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

Author

Paul Curtz, Sep 13 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001045, A036563.

Programs

GAP

List([0..25], n -> 4^(n+1)-3); # Muniru A Asiru, Feb 20 2018

Magma

[4^(n+1)-3: n in [0..30]]; // Vincenzo Librandi, Aug 08 2011

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 *)

PARI

a(n)=4^(n+1)-3 \\ Charles R Greathouse IV, Oct 07 2015

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

Extensions

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

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