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A194285
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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.
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61
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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
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OFFSET
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1,18
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COMMENTS
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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)
tau...............n.......A194293, tau=(1+sqrt(5))/2
...
Questions for each such triangle:
(1) Which rows are constant?
(2) Maximal number of distinct numbers per row?
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REFERENCES
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Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, pages 23-45.
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LINKS
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Ronald L. Graham, Shen Lin, Chio-Shih Lin, Spectra of numbers, Math. Mag. 51 (1978), 174-176.
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EXAMPLE
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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.
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MATHEMATICA
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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}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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