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 A048578 #62 Jul 13 2025 14:22:47
%S A048578 3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,
%T A048578 131073,262145,524289,1048577,2097153,4194305,8388609,16777217,
%U A048578 33554433,67108865,134217729,268435457,536870913,1073741825,2147483649,4294967297,8589934593,17179869185
%N A048578 Pisot sequence L(3,5).
%C A048578 Lexicographically earliest (when ordered) minimal set of generators for A001969 (numbers with an even number of binary 1's) as a group under A003987(.,.) the XOR operation. - _Peter Munn_, Aug 21 2019
%C A048578 Odd numbers with binary weight = 2. - _David James Sycamore_, Feb 02 2025
%D A048578 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%H A048578 Vincenzo Librandi, <a href="/A048578/b048578.txt">Table of n, a(n) for n = 0..2000</a>
%H A048578 Josef Eschgfäller and Andrea Scarpante, <a href="http://arxiv.org/abs/1603.08500">Dichotomic random number generators</a>, arXiv:1603.08500 [math.CO], 2016.
%H A048578 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A048578 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Group.html">Group</a>.
%H A048578 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>.
%H A048578 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A048578 a(n) = 2^(n+1)+1.
%F A048578 a(n) = 3*a(n-1) - 2*a(n-2).
%F A048578 O.g.f.: (3-4*x)/(1-3*x+2*x^2). - _R. J. Mathar_, Nov 23 2007
%F A048578 E.g.f.: exp(x)*(1 + 2*exp(x)). - _Elmo R. Oliveira_, Dec 06 2024
%t A048578 LinearRecurrence[{3,-2},{3,5},40] (* _Harvey P. Dale_, Sep 10 2017 *)
%o A048578 (Magma) [2^(n+1)+1 : n in [0..40]]; // _Vincenzo Librandi_, Sep 01 2011
%o A048578 (PARI) my(x='x+O('x^99)); Vec(1/(1-x)+2/(1-2*x)) \\ _Altug Alkan_, Mar 29 2016
%Y A048578 Subsequence of A000051.
%Y A048578 See A008776 for definitions of Pisot sequences.
%Y A048578 Cf. A001969, A003987.
%Y A048578 Essentially the same as A020737 and A000051.
%K A048578 nonn,easy
%O A048578 0,1
%A A048578 _David W. Wilson_