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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
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 21 2011
STATUS
approved