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A194397 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(14) and < > denotes fractional part.

%I A194397 #15 Feb 15 2021 02:18:51
%S A194397 3,7,11,15,19,23,27,61,65,69,73,77,81,85,89,91,92,93,95,96,97,99,100,
%T A194397 101,103,104,105,107,108,109,111,112,113,115,116,117,119,123,127,131,
%U A194397 135,139,143,147,181,185,189,193,197,201,205,209,211,212,213,215
%N A194397 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) > 0, where r=sqrt(14) and < > denotes fractional part.
%C A194397 See A194368.
%H A194397 Robert Israel, <a href="/A194397/b194397.txt">Table of n, a(n) for n = 1..10000</a>
%p A194397 r:= sqrt(14):
%p A194397 X:= 0: R:= NULL: count:= 0:
%p A194397 for n from 1 while count < 100 do
%p A194397   X:= X + frac(1/2+n*r) - frac(n*r);
%p A194397   if X > 0 then
%p A194397     count:= count+1;
%p A194397     R:= R, n
%p A194397   fi
%p A194397 od:
%p A194397 R; # _Robert Israel_, Nov 25 2020
%t A194397 r = Sqrt[14]; c = 1/2;
%t A194397 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194397 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194397 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194397 Flatten[Position[t1, 1]]       (* A194395 *)
%t A194397 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
%t A194397 Flatten[Position[t2, 1]]       (* A194396 *)
%t A194397 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
%t A194397 Flatten[Position[t3, 1]]       (* A194397 *)
%Y A194397 Cf. A010471 (sqrt(14)), A194368, A194396, A194397.
%K A194397 nonn
%O A194397 1,1
%A A194397 _Clark Kimberling_, Aug 23 2011