



2, 4, 8, 15, 17, 19, 21, 26, 28, 32, 35, 39, 41, 46, 48, 52, 59, 61, 63, 65, 70, 72, 76, 78, 83, 85, 89, 92, 96, 100, 102, 107, 109, 113, 116, 120, 122, 127, 129, 133, 140, 144, 146, 151, 153, 157, 159, 164, 166, 170, 173, 177, 181, 184, 188, 190, 195, 197, 201, 203, 208, 210, 212, 214, 221
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OFFSET

1,1


COMMENTS

Let tau be the tribonacci constant (A058265). Although 1/tau + 1/tau^2 + 1/tau^3 = 1, by Uspensky's 1927 theorem, the three sequences floor(n*tau) (A158919), floor(n*tau^2) (A277722), and floor(n*tau^3) (A277723) cannot form a partition of the nonnegative integers. (Compare Beatty's theorem.)
Entries A277724A277727 investigate how these three sequences meet, and the present sequence gives the numbers not in any of the three sequences. Any two of the three sequences have a nontrivial intersection, while the intersection of all three is {0}.
On the other hand, the three sequences A003144, A003145, A003146, which arise from the tribonacci word, DO form a partition of the positive integers and are closely connected with the three sequences mentioned in the definition.
It would be nice to have bfiles for all the sequences mentioned here. (Many do, but some do not.)


LINKS

JeanFrançois Alcover, Table of n, a(n) for n = 1..2955
S. Beatty, A. Ostrowski, J. Hyslop, and A. C. Aitken, Problems and Solutions: Solutions: 3177, Amer. Math. Monthly, 34 (1927), pp. 159160.
R. L. Graham, On a theorem of Uspensky, Amer. Math. Mnthly, 70 (1963): 407409.
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
J. V. Uspensky, On a problem arising out of the theory of a certain game, Amer. Math. Mnthly., 34 (1927), 516521.


MATHEMATICA

maxTerm = 10000;
a19[n_] := Floor[n*Root[#^3  #^2  #  1&, 1]];
a22[n_] := Floor[n*Root[#^3  3 #^2  #  1&, 1]];
a23[n_] := Floor[n*Root[#^3  7 #^2 + 5 #  1&, 1]];
A19 = Reap[k = 1; While[a19[k] <= maxTerm, Sow[a19[k++]]]][[2, 1]];
A22 = Reap[k = 1; While[a22[k] <= maxTerm, Sow[a22[k++]]]][[2, 1]];
A23 = Reap[k = 1; While[a23[k] <= maxTerm, Sow[a23[k++]]]][[2, 1]];
Select[Range[maxTerm], FreeQ[A19, #] && FreeQ[A22, #] && FreeQ[A23, #]&] (* JeanFrançois Alcover, Dec 06 2018 *)


CROSSREFS

Cf. A003144, A003145, A003146, A058265, A158919, A184820, A277722, A277723, A277724, A277725, A277726, A277727.
Sequence in context: A070008 A033623 A094398 * A287163 A290652 A288313
Adjacent sequences: A277725 A277726 A277727 * A277729 A277730 A277731


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Oct 30 2016


STATUS

approved



