A079044 - OEIS

%I #29 Feb 14 2021 13:00:37

%S 19,39,59,78,98,118,138,157,177,197,217,236,256,276,296,315,335,355,

%T 375,394,414,434,454,473,493,513,532,552,572,592,611,631,651,671,690,

%U 710,730,750,769,789,809,829,848,868,888,908,927,947,967,986

%N Numbers k such that Sum_{j=0..k} sin(j/Pi) < 0.

%F a(n) is asymptotic to c*n where c=19.7...

%F Conjecture: a(n) = floor( 2 * Pi^2 * n ), checked for n <= 10^4. - _Vincenzo Librandi_, Sep 03 2015

%F From _Vaclav Kotesovec_, Feb 15 2019: (Start)

%F Numbers k such that sin(k/(2*Pi)) * sin((k+1)/(2*Pi)) < 0.

%F Numbers k such that cos((2*k+1)/(2*Pi)) > cos(1/(2*Pi)).

%F Numbers k such that k+1 > 2*Pi^2*(floor(k/(2*Pi^2))+1).

%F Numbers k such that k mod (2*Pi^2) > 2*Pi^2 - 1.

%F (End)

%t Select[Range[1, 1000], Sum[Sin[k/Pi], {k,0,#}] < 0&] (* _Vaclav Kotesovec_, Feb 15 2019 *)

%t Select[Range[1, 1000], Cos[(2*# + 1)/(2*Pi)] > Cos[1/(2*Pi)]&] (* _Vaclav Kotesovec_, Feb 15 2019 *)

%t Select[Range[1, 1000], Mod[(2*# + 1)/(2*Pi), 2*Pi] < 1/(2*Pi) || Mod[(2*# + 1)/(2*Pi), 2*Pi] > 2*Pi - 1/(2*Pi) &] (* _Vaclav Kotesovec_, Feb 15 2019 *)

%t Select[Range[1, 1000], # + 1 > 2*Pi^2*(Floor[#/(2*Pi^2)] + 1) &] (* _Vaclav Kotesovec_, Feb 15 2019 *)

%t Select[Range[1, 1000], Mod[#,2*Pi^2] > 2*Pi^2 - 1 &] (* _Vaclav Kotesovec_, Feb 15 2019 *)

%o (PARI) isok(n) = sum(k=0,n,sin(k/Pi)) < 0; \\ _Michel Marcus_, Nov 30 2013

%K nonn

%O 1,1

%A _Benoit Cloitre_, Feb 01 2003