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

0,5

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

Table of n, a(n) for n=0..73.

Wikipedia, Equidistribution theorem.

Formula

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

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
*)

Prog

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

Crossrefs

Cf. A000796, A022844, A336018, A331008.

Sequence in context: A226609 A370730 A157260 * A291486 A177870 A094872

Adjacent sequences: A336014 A336015 A336016 * A336018 A336019 A336020

Keyword

nonn

Author

Andres Cicuttin, Jul 04 2020

Status

approved