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A194368 Numbers n such that Sum_{k=1..n} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(2) and < > denotes fractional part. 68
2, 4, 12, 14, 16, 24, 26, 28, 70, 72, 74, 82, 84, 86, 94, 96, 98, 140, 142, 144, 152, 154, 156, 164, 166, 168, 408, 410, 412, 420, 422, 424, 432, 434, 436, 478, 480, 482, 490, 492, 494, 502, 504, 506, 548, 550, 552, 560, 562, 564, 572, 574, 576, 816, 818 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Suppose that r and c are real numbers, 0 < c < 1, and

...

       s(n) = Sum_{k=1..n} (<c+k*r> - <k*r>)

...

where < > denotes fractional part. The inequalities s(n)<0, s(n)=0, s(n)>0 yield up to three sequences that partition the set of positive integers, as in the examples cited below. Of particular interest are choices of r and c for which s(n)>=0 for every n>=1.

...

Note that s(n) = n*c - Sum_{k=1..n} floor(c + <k*r>). This shows that if c is a rational number p/q, then the range of s(n) is a set of rational numbers having denominator q. In this case, it is easy to prove that if s(n)=0, then n is an integer multiple of q, yielding a sequence of quotients denoted by [[n/q>]] in the following list:

...

r..........p/q....s(n)<0....s(n)=0....[[n/q]]...s(n)>0

sqrt(2)....1/2....(empty)...A194368...A194369...A194370

sqrt(3)....1/2....A194371...A194372.............A194373

sqrt(5)....1/2....(empty)...A194374.............A194375

sqrt(6)....1/2....(empty)...A194376.............A194377

sqrt(7)....1/2....A194378...A194379.............A194380

sqrt(8)....1/2....A194381...A194382...A194383...A194384

sqrt(10)...1/2....(empty)...A194385.............A194386

sqrt(11)...1/2....A194387...A194388.............A194389

sqrt(12)...1/2....(empty)...A194390.............A194391

sqrt(13)...1/2....A194392...A194393.............A194394

sqrt(14)...1/2....A194395...A194396.............A194397

sqrt(15)...1/2....A194398...A194399.............A194400

tau........1/2....A194401...A194402...A194403...A194404

e..........1/2....A194405...A194406.............A194407

pi.........1/2....A194408...A194409.............A194410

sqrt(2)....1/3....A194411...A194412...A194413...A194414

sqrt(3)....1/3....A194415...A194416...A194417...A194418

sqrt(5)....1/3....A194419...A194420.............A194421

sqrt(2)....2/3....A194422...A194423...A194424...A194425

tau.....<tau>/2...A194461.......................A194462

tau.....<tau/2>...A194463.......................A194464

sqrt(2)....1/r.......A194465....................A194466

sqrt(3)....1/r.......A194467....................A194468

...

Next, suppose that r and c are chosen so that s(n)=0 for all n. Then the sets X={n : s(n)<0} and Y={n : s(n)>0} represent a pair of "generalized Beatty sequences" in this sense: if c=1/<r>, the sets X and Y represent the Beatty sequences of 1/<r> and 1<-r>. Examples:

...

r..........c.........X.........Y......

sqrt(2)....r-1.......A003151...A003152

sqrt(3)....r-1.......A003511...A003512

tau........r-1.......A000201...A001950

sqrt(1/2)..r.........A001951...A001952

e..........e-2.......A000062...A098005

REFERENCES

Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963.

LINKS

Table of n, a(n) for n=1..55.

Ronald L. Graham, Shen Lin, Chio-Shih Lin, Spectra of numbers, Math. Mag. 51 (1978), 174-176.

MATHEMATICA

r = Sqrt[2]; c = 1/2;

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

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

t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];

Flatten[Position[t1, 1]] (* empty *)

t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];

Flatten[Position[t2, 1]] (* A194368 *)

%/2 (* A194369 *)

t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];

Flatten[Position[t3, 1]] (* A194370 *)

CROSSREFS

Cf. A184369=(1/2)A184368, A194285, A194469, A194470.

Sequence in context: A087796 A268494 A039587 * A111069 A107295 A039564

Adjacent sequences:  A194365 A194366 A194367 * A194369 A194370 A194371

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 23 2011

STATUS

approved

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Last modified October 24 05:20 EDT 2017. Contains 293835 sequences.