

A336017


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


1



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
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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[[j1]], 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: A256301 A226609 A157260 * A291486 A177870 A094872
Adjacent sequences: A336014 A336015 A336016 * A336018 A336019 A336020


KEYWORD

nonn


AUTHOR

Andres Cicuttin, Jul 04 2020


STATUS

approved



