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%I #40 Sep 08 2022 08:44:57
%S 3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,
%T 131073,262145,524289,1048577,2097153,4194305,8388609,16777217,
%U 33554433,67108865,134217729,268435457,536870913,1073741825,2147483649,4294967297,8589934593
%N Pisot sequence L(3,5).
%C 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
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%H Vincenzo Librandi, <a href="/A048578/b048578.txt">Table of n, a(n) for n = 0..2000</a>
%H Josef Eschgfäller, Andrea Scarpante, <a href="http://arxiv.org/abs/1603.08500">Dichotomic random number generators</a>, arXiv:1603.08500 [math.CO], 2016.
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Group.html">Group</a>
%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F a(n) = 2^(n+1)+1.
%F a(n) = 3*a(n-1) - 2*a(n-2).
%F O.g.f.: (3-4*x)/(1-3*x+2*x^2). - _R. J. Mathar_, Nov 23 2007
%t LinearRecurrence[{3,-2},{3,5},40] (* _Harvey P. Dale_, Sep 10 2017 *)
%o (Magma) [2^(n+1)+1 : n in [0..40]]; // _Vincenzo Librandi_, Sep 01 2011
%o (PARI) x='x+O('x^99); Vec(1/(1-x)+2/(1-2*x)) \\ _Altug Alkan_, Mar 29 2016
%Y Subsequence of A000051.
%Y See A008776 for definitions of Pisot sequences.
%Y Cf. A001969, A003987.
%K nonn,easy
%O 0,1
%A _David W. Wilson_
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