A013942 Triangle of numbers T(n,k) = floor(2n/k), k=1..2n, read by rows.

2, 1, 4, 2, 1, 1, 6, 3, 2, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 14, 7, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 20, 10, 6, 5, 4, 3

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabf
• easy
• nice

Comments

a(n) is also the leading term in period of continued fraction for n-th nonsquare.

Row A026741(n) contains n and all rows with a smaller row number do not contain n. - Reinhard Zumkeller, Jun 04 2013

Examples

			First four rows:
  2 1
  4 2 1 1
  6 3 2 1 1 1
  8 4 2 2 1 1 1 1
  ...
		

Links

• Reinhard Zumkeller, Rows n = 1..100 of table, flattened

Crossrefs

Cf. A010766.

Cf. A005843 (row lengths and left edge), A062550 (row sums).

Programs

Haskell

a013942 n k = a013942_tabf !! (n-1) !! (k-1)
a013942_row n = map (div (n * 2)) [1 .. 2 * n]
a013942_tabf = map a013942_row [1 ..]
-- Reinhard Zumkeller, Jun 04 2013

Mathematica

f[n_,h_]:=FractionalPart[(n^2+h)^(1/2)];
g[n_,h_]:=Floor[1/f[n,h]];
TableForm[Table[g[n,h],{n,1,13},{h,1,2n}]]

PARI

T(n, k) = 2*n\k;
tabf(nn) = for (n=1, nn, for (k=1, 2*n, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Sep 30 2016

Formula

T(n,k) = floor(2n/k), k=1,...,2n.

T(n,k) = [1/{sqrt(k+n^2)}], k=1,2,...,2n, {}=fractional part, []=floor.

Extensions

Keyword tabl replaced by tabf and missing a(90)=1 inserted by Reinhard Zumkeller, Jun 04 2013