A036414 Number of empty intervals when fractional_part(i*phi) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, 0, 2, 3, 1, 2, 0, 3, 2, 4, 3, 1, 3, 3, 4, 3, 2, 4, 5, 0, 4, 5, 4, 8, 6, 6, 5, 2, 5, 5, 5, 5, 8, 5, 5, 4, 8, 6, 6, 5, 0, 6, 7, 8, 7, 6, 8, 8, 11, 9, 8, 10, 9, 4, 9, 9, 9, 8, 8, 9, 8, 12, 8, 8, 10, 9, 6, 9, 8, 11, 10, 8, 10, 10, 0, 10, 11, 9, 12, 12, 14

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• look

References

• H. Steinhaus, Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, pp. 48-49, 1983.

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Equidistributed Sequence
• Eric Weisstein's World of Mathematics, Golden Ratio

Crossrefs

Cf. A036415 (positions of 0).

Cf. similar sequences with other constants: A036412 (e), A036416 (Pi), A046157 (gamma).

Programs

Mathematica

Table[Length@Complement[Range[n] - 1, Floor[n*FractionalPart[GoldenRatio*Range[n]]]], {n, 95}] (* Ivan Neretin, Jan 23 2018 *)
Table[Count[BinCounts[FractionalPart[GoldenRatio Range[n]], {0, 1, 1/n}], 0], {n, 95}] (* Eric W. Weisstein, Apr 17 2024 *)