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Number of empty intervals when fractional_part(i*phi) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.
5

%I #20 Apr 17 2024 20:45:04

%S 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,

%T 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,

%U 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

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

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

%H Ivan Neretin, <a href="/A036414/b036414.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EquidistributedSequence.html">Equidistributed Sequence</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>

%t Table[Length@Complement[Range[n] - 1, Floor[n*FractionalPart[GoldenRatio*Range[n]]]], {n, 95}] (* _Ivan Neretin_, Jan 23 2018 *)

%t Table[Count[BinCounts[FractionalPart[GoldenRatio Range[n]], {0, 1, 1/n}], 0], {n, 95}] (* _Eric W. Weisstein_, Apr 17 2024 *)

%Y Cf. A036415 (positions of 0).

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

%K nonn,look

%O 1,9

%A _Eric W. Weisstein_