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Numbers m such that Sum_{k=1..m} (<c + k*r> - <k*r>) > 0, where r=(1+sqrt(5))/2 and c=(-1+sqrt(5))/4, and < > denotes fractional part.
1

%I #9 Feb 14 2021 21:40:41

%S 1,2,4,5,7,10,13,14,15,17,18,20,23,26,27,28,30,31,33,34,35,36,37,38,

%T 39,40,41,43,44,46,47,48,49,50,51,52,53,54,56,57,59,60,62,65,68,69,70,

%U 72,73,75,78,81,82,83,85,86,88,89,90,91,92,93,94,95,96,98,99,101

%N Numbers m such that Sum_{k=1..m} (<c + k*r> - <k*r>) > 0, where r=(1+sqrt(5))/2 and c=(-1+sqrt(5))/4, and < > denotes fractional part.

%C See A194368.

%t r = GoldenRatio; c = (1/2) FractionalPart[r];

%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]] (* A184461 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t3, 1]] (* A184462 *)

%Y Cf. A194368.

%K nonn

%O 1,2

%A _Clark Kimberling_, Aug 24 2011