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%I #35 Mar 07 2021 01:04:51
%S 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,
%T 7,12,16,22,27,33,3,8,14,20,26,33,39,3,10,16,23,30,38,45,3,11,18,26,
%U 34,43,52,4,12,20,29,38,48,57,3,13,22,32,42,53,63,3,14,24
%N a(n) = floor(frac(Pi*n)*n), where frac denotes the fractional part.
%C 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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Equidistribution_theorem">Equidistribution theorem</a>.
%F a(n) = floor((Pi*n - floor(Pi*n))*n).
%t a[n_]:=Floor[FractionalPart[Pi*n]*n];
%t Table[a[n], {n, 0, 100}]
%t (* uncomment following lines to count increasing subsequences.
%t The function MySplit[c] splits the sequence c into monotonically increasing subsequences *)
%t (*
%t MySplit[c_List]:=Module[{d={{c[[1]]}},k=1},
%t Do[If[c[[j]]>c[[j-1]],AppendTo[d[[k]],c[[j]]] ,AppendTo[d,{c[[j]]}];k++],{j,2,Length[c]}];Return[d]];
%t tab=Table[a[n], {n, 1, 2^20 }];
%t Map[Length, MySplit[tab], 1] // Tally
%t *)
%o (PARI) a(n) = frac(Pi*n)*n\1; \\ _Michel Marcus_, Jul 07 2020
%Y Cf. A000796, A022844, A336018, A331008.
%K nonn
%O 0,5
%A _Andres Cicuttin_, Jul 04 2020
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