cp's OEIS Frontend

A194377 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(6) and < > denotes fractional part.

%I A194377 #12 Feb 15 2021 02:20:20
%S A194377 1,3,5,7,9,10,11,12,13,14,15,16,17,18,19,21,23,25,27,29,30,31,32,33,
%T A194377 34,35,36,37,38,39,41,43,45,47,49,50,51,52,53,54,55,56,57,58,59,61,63,
%U A194377 65,67,69,70,71,72,73,74,75,76,77,78,79,81,83,85,87,89,90,91,92,93,94,95,96,97,98,99,100
%N A194377 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) > 0, where r=sqrt(6) and < > denotes fractional part.
%C A194377 See A194368. Although a(n)=A007957(n) for n = 1..70, the number 208, for example, is here but not A007957.
%t A194377 r = Sqrt[6]; c = 1/2;
%t A194377 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194377 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194377 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 500}];
%t A194377 Flatten[Position[t1, 1]]   (* empty *)
%t A194377 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
%t A194377 Flatten[Position[t2, 1]]   (* A194376 *)
%t A194377 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
%t A194377 Flatten[Position[t3, 1]]   (* A194377 *)
%o A194377 (PARI) is(n)=my(r=sqrt(6),f=x->x-x\1);sum(k=1,n,f(1/2+k*r)-f(k*r))>0 \\ _Charles R Greathouse IV_, Jul 25 2012
%Y A194377 Cf. A007957, A194368, A194376.
%K A194377 nonn
%O A194377 1,2
%A A194377 _Clark Kimberling_, Aug 23 2011