login
Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) < 0, where r=sqrt(2) and < > denotes fractional part.
5

%I #13 Feb 15 2021 02:19:28

%S 2,14,26,38,41,43,44,50,53,55,56,62,65,67,68,70,71,72,73,74,77,79,80,

%T 82,83,84,85,86,89,91,92,94,95,96,97,98,101,113,125,137,140,142,143,

%U 149,152,154,155,161,164,166,167,212,224,236,239,241,242,248,251

%N Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) < 0, where r=sqrt(2) and < > denotes fractional part.

%C See A194368.

%H G. C. Greubel, <a href="/A194411/b194411.txt">Table of n, a(n) for n = 1..1000</a>

%t r = Sqrt[2]; c = 1/3;

%t x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]

%t y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]

%t t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t1, 1]] (* A194411 *)

%t t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];

%t Flatten[Position[t2, 1]] (* A194412 *)

%t %/3 (* A194413 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];

%t Flatten[Position[t3, 1]] (* A194414 *)

%Y Cf. A002193, A194368, A194412, A194413, A194414.

%K nonn

%O 1,1

%A _Clark Kimberling_, Aug 24 2011