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.

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, 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, 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

Offset: 1

Author

Clark Kimberling, Aug 21 2011

Keywords

• nonn
• tabl

Comments

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:

...

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

sqrt(2)...........n.......A194285

sqrt(2)...........2n......A194286

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

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

sqrt(3)...........n.......A194289

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

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

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

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

tau...............2n......A194294

tau...............n^2.....A194295

tau...............2^n.....A194296

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

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

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

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

sqrt(5)...........n.......A194301

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

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

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

pi................n.......A194305

pi................2n......A194306

pi................n^2.....A194307

pi................2^n.....A194308

e.................n.......A194309

e.................2n......A194310

e.................n^2.....A194311

e.................2^n.....A194312

sqrt(6)...........n.......A194313

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

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

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

sqrt(8)...........n.......A194317

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2-tau.............n.......A194333

2-tau.............2n......A194334

2-tau.............n^2.....A194335

2-tau.............2^n.....A194336

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

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

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

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

3-e...............n.......A194341

3-e...............2n......A194342

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

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

...

Questions for each such triangle:

(1) Which rows are constant?

(2) Maximal number of distinct numbers per row?

Examples

			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..1..1
Take n=6, r=sqrt(2):
(r)=-1+r=0.41412... in [2/6,3/6)
(2r)=-2+2r=0.828... in [4/6,5/6)
(3r)=-4+3r=0.242... in [1/6,2/6)
(4r)=-5+4r=0.656... in [3/6,4/6)
(5r)=-7+5r=0.071... in [0/6,1/6)
(6r)=-8+6r=0.485... in [2/6,3/6),
so that row 6 is 1..1..2..1..1..0.
		

References

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

Links

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

Crossrefs

Cf. A194286, A194287, A194288.

Programs

Mathematica

r = Sqrt[2];
f[n_, k_, i_] := If[(k - 1)/n <= FractionalPart[i*r] < k/n, 1, 0]
g[n_, k_] := Sum[f[n, k, i], {i, 1, n}]
TableForm[Table[g[n, k], {n, 1, 20}, {k, 1, n}]]
Flatten[%]  (* A194285 *)