This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A141725 #60 Nov 15 2023 15:21:02
%S A141725 1,13,61,253,1021,4093,16381,65533,262141,1048573,4194301,16777213,
%T A141725 67108861,268435453,1073741821,4294967293,17179869181,68719476733,
%U A141725 274877906941,1099511627773,4398046511101,17592186044413,70368744177661
%N A141725 a(n) = 4^(n+1) - 3.
%C A141725 Inverse binomial transform yields A003946 with A003946(1)=4 deleted. - _R. J. Mathar_, Sep 13 2008
%C A141725 Starting with n=1, binary numbers of the form 1X01 where X is an odd number of 1's. - _Brad Clardy_, Mar 22 2011
%C A141725 Column 4 of A193871. - _Omar E. Pol_, Aug 22 2011
%C A141725 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
%H A141725 Vincenzo Librandi, <a href="/A141725/b141725.txt">Table of n, a(n) for n = 0..1000</a>
%H A141725 John Elias, <a href="/A141725/a141725.png">Illustration of Initial Terms: Sierpinski Square Curve</a>
%H A141725 Mattia Fregola, <a href="https://docs.google.com/spreadsheets/d/1629UXZ07lVK1-LVR0T7u1RDVKeW4f55K688CAtS5maw/edit?usp=sharing">Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450</a>
%H A141725 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F A141725 a(n) = 10*A001045(2*n) + A001045(2*n+1).
%F A141725 a(n) = 4*a(n-1) + 9 for n > 0, a(0) = 1.
%F A141725 a(n) = A036563(2*n+2).
%F A141725 G.f.: (1 + 8*x)/((1 - x)*(1 - 4*x)). - _R. J. Mathar_, Sep 13 2008
%F A141725 a(n) = 4^n - 3, with offset 1. - _Omar E. Pol_, Aug 22 2011
%F A141725 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
%F A141725 E.g.f.: exp(4*x) - 3*exp(x). - _Elmo R. Oliveira_, Nov 15 2023
%p A141725 a:= n-> 4^(n+1)-3: seq(a(n), n=0..25); # _Muniru A Asiru_, Feb 20 2018
%t A141725 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 *)
%o A141725 (Magma) [4^(n+1)-3: n in [0..30]]; // _Vincenzo Librandi_, Aug 08 2011
%o A141725 (PARI) a(n)=4^(n+1)-3 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A141725 (GAP) List([0..25], n -> 4^(n+1)-3); # _Muniru A Asiru_, Feb 20 2018
%Y A141725 Cf. A001045, A036563.
%K A141725 nonn,easy
%O A141725 0,2
%A A141725 _Paul Curtz_, Sep 13 2008
%E A141725 Edited by _N. J. A. Sloane_, Sep 13 2008
%E A141725 More terms from _R. J. Mathar_, Sep 13 2008