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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
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,18 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? 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. EXAMPLE 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. 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 *) CROSSREFS Cf. A194286, A194287, A194288. Sequence in context: A279907 A225654 A236747 * A135341 A033665 A104234 Adjacent sequences:  A194282 A194283 A194284 * A194286 A194287 A194288 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Aug 21 2011 STATUS approved

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Last modified December 18 22:08 EST 2018. Contains 318245 sequences. (Running on oeis4.)