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 A153893 #57 Jul 18 2025 09:33:56
%S A153893 2,5,11,23,47,95,191,383,767,1535,3071,6143,12287,24575,49151,98303,
%T A153893 196607,393215,786431,1572863,3145727,6291455,12582911,25165823,
%U A153893 50331647,100663295,201326591,402653183,805306367,1610612735,3221225471
%N A153893 a(n) = 3*2^n - 1.
%C A153893 A020944(a(n)) = 0. - _Reinhard Zumkeller_, Mar 13 2011
%C A153893 a(n) + a(n-1)^2 is a perfect square. - _Vincenzo Librandi_, Oct 28 2011
%C A153893 Number of distinct continued fractions of n terms chosen from {1,2}. - _Clark Kimberling_, Jul 20 2015
%C A153893 Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - _Robert Price_, Mar 09 2017
%C A153893 This sequence has been used by the ninth-century mathematician Thabit ibn Qurra to devise the first method to construct amicable pairs (see Tattersall). - _Stefano Spezia_, Jul 18 2025
%D A153893 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 138.
%H A153893 Vincenzo Librandi, <a href="/A153893/b153893.txt">Table of n, a(n) for n = 0..1000</a>
%H A153893 Gennady Eremin, <a href="https://arxiv.org/abs/2405.16143">Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant</a>, arXiv:2405.16143 [math.CO], 2024. See pp. 2-5, 14.
%H A153893 Gennady Eremin, <a href="https://arxiv.org/abs/2501.17090">Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers</a>, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
%H A153893 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A153893 a(n) = a(n-1)*2 + 1, a(0)=2.
%F A153893 a(n) = A083329(n+1).
%F A153893 a(n) = A055010(n+1).
%F A153893 G.f.: (2 - x)/((1-x)(1-2x)). - _R. J. Mathar_, Feb 13 2009
%F A153893 a(n) = A083416(2n) = A033484(n) + 1. - _Philippe Deléham_, Apr 14 2013
%F A153893 From _G. C. Greubel_, Sep 01 2016: (Start)
%F A153893 a(n) = 3*a(n-1) - 2*a(n-2).
%F A153893 E.g.f.: 3*exp(2*x) - exp(x). (End)
%t A153893 Table[3*2^n - 1 , {n,0,25}] (* _G. C. Greubel_, Sep 01 2016 *)
%t A153893 LinearRecurrence[{3,-2},{2,5},40] (* _Harvey P. Dale_, Mar 01 2024 *)
%o A153893 (Magma) [3*2^n-1: n in [0..30]]; // _Vincenzo Librandi_, Oct 28 2011
%o A153893 (PARI) a(n)=3*2^n-1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A153893 Cf. A283508.
%K A153893 nonn,easy
%O A153893 0,1
%A A153893 _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009
%E A153893 Edited by _N. J. A. Sloane_, Feb 14 2009