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A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).

%I A158919 #55 Sep 08 2022 08:45:43
%S A158919 0,1,3,5,7,9,11,12,14,16,18,20,22,23,25,27,29,31,33,34,36,38,40,42,44,
%T A158919 45,47,49,51,53,55,57,58,60,62,64,66,68,69,71,73,75,77,79,80,82,84,86,
%U A158919 88,90,91,93,95,97,99,101,103,104,106,108,110
%N A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).
%C A158919 Also called the spectrum of tau. Note that A276384 agrees with this sequence for n <= 17160 but disagrees beyond that point. In fact a(17161) = 31564, whereas A276384(17161) = 31563. - _N. J. A. Sloane_, Sep 03 2016
%H A158919 N. J. A. Sloane, <a href="/A158919/b158919.txt">Table of n, a(n) for n = 0..20000</a>
%H A158919 A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, <a href="https://arxiv.org/abs/1809.08690">Almost Beatty Partitions</a>, arXiv:1809.08690 [math.NT], 2018.
%H A158919 Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%H A158919 N. J. A. Sloane, <a href="/A158919/a158919.txt">Table of n, a(n) for n = 0..100000</a>
%H A158919 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A158919 a(n) = floor(n*A058265).
%e A158919 a(3) = floor(3*q) = floor(3*1.8392867...) = floor(5.51786...) = 5.
%p A158919 x := (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)) ;
%p A158919 [seq(floor(n*x),n=0..200)]; # _R. J. Mathar_, Sep 11 2011
%t A158919 a[n_] := Floor[n Root[#^3 - #^2 - # - 1&, 1]];
%t A158919 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 28 2018 *)
%o A158919 (Magma) [Floor(n * (1/3 + 1/3 * (19 - 3 * Sqrt(33))^(1/3) + 1/3 * (19 + 3 * Sqrt(33))^(1/3))) : n in [0..80]]; // _Vincenzo Librandi_, Oct 28 2018
%Y A158919 Cf. A058265, A140099 (spectrum of 1+tau), A276384, A277722, A277723.
%K A158919 nonn
%O A158919 0,3
%A A158919 Eric Culver (weux082690(AT)yahoo.com), Mar 30 2009