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 A277724-A277727 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 b-files for all the sequences mentioned here. (Many do, but some do not.)

LINKS

Jean-François 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. 159-160.

R. L. Graham, On a theorem of Uspensky, Amer. Math. Mnthly, 70 (1963): 407-409.

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), 516-521.

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, #]&] (* Jean-François Alcover, Dec 06 2018 *)

CROSSREFS

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 30 2016

STATUS

approved