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 A194368 Numbers m such that Sum_{k=1..m} (<1/2 + 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(m) = Sum_{k=1..m} ( - ) ... where < > denotes fractional part. The inequalities s(m) < 0, s(m) = 0, s(m) > 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(m) >= 0 for every m >= 1. . Note that s(m) = m*c - Sum_{k=1..m} floor(c + ). This shows that if c is a rational number p/q, then the range of s(m) is a set of rational numbers having denominator q. In this case, it is easy to prove that if s(m)=0, then m is an integer multiple of q, yielding a sequence of quotients denoted by [[m/q>]] in the following list: . r..........p/q....s(m)<0....s(m)=0....[[m/q]]...s(m)>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...../2...A194461.......................A194462 tau........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(m)=0 for all m. Then the sets X={m : s(m)<0} and Y={m : s(m)>0} represent a pair of "generalized Beatty sequences" in this sense: if c=1/, the sets X and Y represent the Beatty sequences of 1/ 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: A306851 A316634 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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