A194335 Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=n^2, 1<=k<=n, r=2-tau, where tau=(1+sqrt(5))/2, the golden ratio.

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 5, 4, 5, 6, 7, 5, 7, 6, 6, 7, 8, 7, 6, 8, 7, 8, 8, 8, 9, 7, 8, 8, 8, 8, 10, 8, 10, 9, 9, 9, 9, 9, 10, 10, 11, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 11, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13

Offset: 1

Author

Clark Kimberling, Aug 22 2011

Keywords

• nonn
• tabl

Comments

See A194285.

Examples

			First eight rows:
1
2..2
3..3..3
4..4..4..4
5..5..6..5..4
5..6..7..5..7..6
6..7..8..7..6..8..7
8..8..8..9..7..8..8..8
		

Crossrefs

Cf. A194285.

Programs

Mathematica

r = 2-GoldenRatio;
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^2}]
TableForm[Table[g[n, k], {n, 1, 14}, {k, 1, n}]]
Flatten[%]    (* A194335 *)