A336017 a(n) = floor(frac(Pi*n)*n), where frac denotes the fractional part.

0, 0, 0, 1, 2, 3, 5, 6, 1, 2, 4, 6, 8, 10, 13, 1, 4, 6, 9, 13, 16, 20, 2, 5, 9, 13, 17, 22, 27, 3, 7, 12, 16, 22, 27, 33, 3, 8, 14, 20, 26, 33, 39, 3, 10, 16, 23, 30, 38, 45, 3, 11, 18, 26, 34, 43, 52, 4, 12, 20, 29, 38, 48, 57, 3, 13, 22, 32, 42, 53, 63, 3, 14, 24

Offset: 0

Author

Andres Cicuttin, Jul 04 2020

Keywords

• nonn

Comments

It seems that the sequence can be split into consecutive short monotonically increasing subsequences. For example, the first 2^20 terms can be split into 139188 subsequences of 7 terms and 9281 subsequences of 8 terms (see commented part of Mathematica program). The distance between two consecutive terms, a(k) and a(k+1), of the same increasing subsequence is about k/7.

Links

• Wikipedia, Equidistribution theorem.

Crossrefs

Cf. A000796, A022844, A336018, A331008.

Programs

Mathematica

a[n_]:=Floor[FractionalPart[Pi*n]*n];
Table[a[n], {n, 0, 100}]
(* uncomment following lines to count increasing subsequences.
The function MySplit[c] splits the sequence c into monotonically increasing subsequences *)
(*
MySplit[c_List]:=Module[{d={{c[[1]]}},k=1},
Do[If[c[[j]]>c[[j-1]],AppendTo[d[[k]],c[[j]]] ,AppendTo[d,{c[[j]]}];k++],{j,2,Length[c]}];Return[d]];
tab=Table[a[n], {n, 1, 2^20 }];
Map[Length, MySplit[tab], 1] // Tally
*)

PARI

a(n) = frac(Pi*n)*n\1; \\ Michel Marcus, Jul 07 2020

Formula

a(n) = floor((Pi*n - floor(Pi*n))*n).