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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.

%I A194285 #11 Sep 26 2017 11:14:51
%S A194285 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 A194285 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 A194285 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 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.
%C A194285 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 A194285 ...
%C A194285 r.................s(n)....g(n,k)
%C A194285 sqrt(2)...........n.......A194285
%C A194285 sqrt(2)...........2n......A194286
%C A194285 sqrt(2)...........n^2.....A194287
%C A194285 sqrt(2)...........2^n.....A194288
%C A194285 sqrt(3)...........n.......A194289
%C A194285 sqrt(3)...........2n......A194290
%C A194285 sqrt(3)...........n^2.....A194291
%C A194285 sqrt(3)...........2^n.....A194292
%C A194285 tau...............n.......A194293, tau=(1+sqrt(5))/2
%C A194285 tau...............2n......A194294
%C A194285 tau...............n^2.....A194295
%C A194285 tau...............2^n.....A194296
%C A194285 (-1+sqrt(3))/2....n.......A194297
%C A194285 (-1+sqrt(3))/2....2n......A194298
%C A194285 (-1+sqrt(3))/2....n^2.....A194299
%C A194285 (-1+sqrt(3))/2....2^n.....A194300
%C A194285 sqrt(5)...........n.......A194301
%C A194285 sqrt(5)...........2n......A194302
%C A194285 sqrt(5)...........n^2.....A194303
%C A194285 sqrt(5)...........2^n.....A194304
%C A194285 pi................n.......A194305
%C A194285 pi................2n......A194306
%C A194285 pi................n^2.....A194307
%C A194285 pi................2^n.....A194308
%C A194285 e.................n.......A194309
%C A194285 e.................2n......A194310
%C A194285 e.................n^2.....A194311
%C A194285 e.................2^n.....A194312
%C A194285 sqrt(6)...........n.......A194313
%C A194285 sqrt(6)...........2n......A194314
%C A194285 sqrt(6)...........n^2.....A194315
%C A194285 sqrt(6)...........2^n.....A194316
%C A194285 sqrt(8)...........n.......A194317
%C A194285 sqrt(8)...........2n......A194318
%C A194285 sqrt(8)...........n^2.....A194319
%C A194285 sqrt(8)...........2^n.....A194320
%C A194285 sqrt(1/2).........n.......A194321
%C A194285 sqrt(1/2).........2n......A194322
%C A194285 sqrt(1/2).........n^2.....A194323
%C A194285 sqrt(1/2).........2^n.....A194324
%C A194285 2-sqrt(2).........n.......A194325
%C A194285 2-sqrt(2).........2n......A194326
%C A194285 2-sqrt(2).........n^2.....A194327
%C A194285 2-sqrt(2).........2^n.....A194328
%C A194285 2-sqrt(3).........n.......A194329
%C A194285 2-sqrt(3).........2n......A194330
%C A194285 2-sqrt(3).........n^2.....A194331
%C A194285 2-sqrt(3).........2^n.....A194332
%C A194285 2-tau.............n.......A194333
%C A194285 2-tau.............2n......A194334
%C A194285 2-tau.............n^2.....A194335
%C A194285 2-tau.............2^n.....A194336
%C A194285 3-sqrt(5).........n.......A194337
%C A194285 3-sqrt(5).........2n......A194338
%C A194285 3-sqrt(5).........n^2.....A194339
%C A194285 3-sqrt(5).........2^n.....A194340
%C A194285 3-e...............n.......A194341
%C A194285 3-e...............2n......A194342
%C A194285 3-e...............n^2.....A194343
%C A194285 3-e...............2^n.....A194344
%C A194285 ...
%C A194285 Questions for each such triangle:
%C A194285 (1)  Which rows are constant?
%C A194285 (2)  Maximal number of distinct numbers per row?
%D A194285 Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, pages 23-45.
%H A194285 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 A194285 1
%e A194285 1..1
%e A194285 1..1..1
%e A194285 1..1..1..1
%e A194285 1..1..1..1..1
%e A194285 1..1..2..1..1..0
%e A194285 1..1..1..1..1..1..1
%e A194285 1..1..1..2..0..1..1..1
%e A194285 Take n=6, r=sqrt(2):
%e A194285 (r)=-1+r=0.41412... in [2/6,3/6)
%e A194285 (2r)=-2+2r=0.828... in [4/6,5/6)
%e A194285 (3r)=-4+3r=0.242... in [1/6,2/6)
%e A194285 (4r)=-5+4r=0.656... in [3/6,4/6)
%e A194285 (5r)=-7+5r=0.071... in [0/6,1/6)
%e A194285 (6r)=-8+6r=0.485... in [2/6,3/6),
%e A194285 so that row 6 is 1..1..2..1..1..0.
%t A194285 r = Sqrt[2];
%t A194285 f[n_, k_, i_] := If[(k - 1)/n <= FractionalPart[i*r] < k/n, 1, 0]
%t A194285 g[n_, k_] := Sum[f[n, k, i], {i, 1, n}]
%t A194285 TableForm[Table[g[n, k], {n, 1, 20}, {k, 1, n}]]
%t A194285 Flatten[%]  (* A194285 *)
%Y A194285 Cf. A194286, A194287, A194288.
%K A194285 nonn,tabl
%O A194285 1,18
%A A194285 _Clark Kimberling_, Aug 21 2011