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A194285 Triangular array:  g(n,k)=number of fractional parts (i*sqrt(2)) in interval [(k-1)/n, k/n], for 1<=i<=n, 1<=k<=n. 61

%I

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,0,1,1,1,1,1,1,1,1,1,1,2,0,1,

%T 1,1,1,0,2,1,1,1,1,1,1,1,1,1,1,2,0,1,1,2,0,1,1,1,1,1,1,1,1,1,2,0,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,2,1,1,1,1,1,1,1,1,1,0,1,1,2,1,1,0

%N Triangular array: g(n,k)=number of fractional parts (i*sqrt(2)) in interval [(k-1)/n, k/n], for 1<=i<=n, 1<=k<=n.

%C Row n of A194285 counts fractional parts (i*r), for i=1,2,...,n, in each of the intervals indicated. It is of interest to count (i*r) for i=1,2,...,s(n) for various choices of s(n), such as 2n, n^2, and 2^n. In each case, (n-th row sum)=s(n). Examples:

%C ...

%C r.................s(n)....g(n,k)

%C sqrt(2)...........n.......A194285

%C sqrt(2)...........2n......A194286

%C sqrt(2)...........n^2.....A194287

%C sqrt(2)...........2^n.....A194288

%C sqrt(3)...........n.......A194289

%C sqrt(3)...........2n......A194290

%C sqrt(3)...........n^2.....A194291

%C sqrt(3)...........2^n.....A194292

%C tau...............n.......A194293, tau=(1+sqrt(5))/2

%C tau...............2n......A194294

%C tau...............n^2.....A194295

%C tau...............2^n.....A194296

%C (-1+sqrt(3))/2....n.......A194297

%C (-1+sqrt(3))/2....2n......A194298

%C (-1+sqrt(3))/2....n^2.....A194299

%C (-1+sqrt(3))/2....2^n.....A194300

%C sqrt(5)...........n.......A194301

%C sqrt(5)...........2n......A194302

%C sqrt(5)...........n^2.....A194303

%C sqrt(5)...........2^n.....A194304

%C pi................n.......A194305

%C pi................2n......A194306

%C pi................n^2.....A194307

%C pi................2^n.....A194308

%C e.................n.......A194309

%C e.................2n......A194310

%C e.................n^2.....A194311

%C e.................2^n.....A194312

%C sqrt(6)...........n.......A194313

%C sqrt(6)...........2n......A194314

%C sqrt(6)...........n^2.....A194315

%C sqrt(6)...........2^n.....A194316

%C sqrt(8)...........n.......A194317

%C sqrt(8)...........2n......A194318

%C sqrt(8)...........n^2.....A194319

%C sqrt(8)...........2^n.....A194320

%C sqrt(1/2).........n.......A194321

%C sqrt(1/2).........2n......A194322

%C sqrt(1/2).........n^2.....A194323

%C sqrt(1/2).........2^n.....A194324

%C 2-sqrt(2).........n.......A194325

%C 2-sqrt(2).........2n......A194326

%C 2-sqrt(2).........n^2.....A194327

%C 2-sqrt(2).........2^n.....A194328

%C 2-sqrt(3).........n.......A194329

%C 2-sqrt(3).........2n......A194330

%C 2-sqrt(3).........n^2.....A194331

%C 2-sqrt(3).........2^n.....A194332

%C 2-tau.............n.......A194333

%C 2-tau.............2n......A194334

%C 2-tau.............n^2.....A194335

%C 2-tau.............2^n.....A194336

%C 3-sqrt(5).........n.......A194337

%C 3-sqrt(5).........2n......A194338

%C 3-sqrt(5).........n^2.....A194339

%C 3-sqrt(5).........2^n.....A194340

%C 3-e...............n.......A194341

%C 3-e...............2n......A194342

%C 3-e...............n^2.....A194343

%C 3-e...............2^n.....A194344

%C ...

%C Questions for each such triangle:

%C (1) Which rows are constant?

%C (2) Maximal number of distinct numbers per row?

%D Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, pages 23-45.

%H Ronald L. Graham, Shen Lin, Chio-Shih Lin, <a href="http://www.jstor.org/stable/2689998">Spectra of numbers</a>, Math. Mag. 51 (1978), 174-176.

%e 1

%e 1..1

%e 1..1..1

%e 1..1..1..1

%e 1..1..1..1..1

%e 1..1..2..1..1..0

%e 1..1..1..1..1..1..1

%e 1..1..1..2..0..1..1..1

%e Take n=6, r=sqrt(2):

%e (r)=-1+r=0.41412... in [2/6,3/6)

%e (2r)=-2+2r=0.828... in [4/6,5/6)

%e (3r)=-4+3r=0.242... in [1/6,2/6)

%e (4r)=-5+4r=0.656... in [3/6,4/6)

%e (5r)=-7+5r=0.071... in [0/6,1/6)

%e (6r)=-8+6r=0.485... in [2/6,3/6),

%e so that row 6 is 1..1..2..1..1..0.

%t r = Sqrt[2];

%t f[n_, k_, i_] := If[(k - 1)/n <= FractionalPart[i*r] < k/n, 1, 0]

%t g[n_, k_] := Sum[f[n, k, i], {i, 1, n}]

%t TableForm[Table[g[n, k], {n, 1, 20}, {k, 1, n}]]

%t Flatten[%] (* A194285 *)

%Y Cf. A194286, A194287, A194288.

%K nonn,tabl

%O 1,18

%A _Clark Kimberling_, Aug 21 2011

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Last modified November 15 13:56 EST 2019. Contains 329149 sequences. (Running on oeis4.)