A182210 Triangle T(n,k) = floor(k*(n+1)/(k+1)), 1 <= k <= n.

1, 1, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 5, 3, 4, 5, 5, 5, 6, 4, 5, 6, 6, 6, 6, 7, 4, 6, 6, 7, 7, 7, 7, 8, 5, 6, 7, 8, 8, 8, 8, 8, 9, 5, 7, 8, 8, 9, 9, 9, 9, 9, 10, 6, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 6, 8, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 7, 9, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 7, 10, 11, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 8, 10, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15

Offset: 1

Author

Dennis P. Walsh, Apr 18 2012

Keywords

• nonn
• nice
• easy
• tabl
• look

Comments

T(n,k) is the maximum number of wins in a sequence of n games in which the longest winning streak is of length k.

T(n,k) generalizes the pattern found in sequence A004523 where A004523(n) = floor(2n/3).

Examples

			T(12,4) = 10 since 10 is the maximum number of wins in a 12-game sequence in which the longest winning streak is 4. One such sequence with 10 wins is WWWWLWWWWLWW.
The triangle T(n,k) begins
1,
1, 2,
2, 2, 3,
2, 3, 3,  4,
3, 4, 4,  4,  5,
3, 4, 5,  5,  5,  6,
4, 5, 6,  6,  6,  6,  7,
4, 6, 6,  7,  7,  7,  7,  8,
5, 6, 7,  8,  8,  8,  8,  8,  9,
5, 7, 8,  8,  9,  9,  9,  9,  9, 10,
6, 8, 9,  9, 10, 10, 10, 10, 10, 10, 11,
6, 8, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12,
		

Links

• Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
• Sela Fried and Toufik Mansour, The total number of descents and levels in (cyclic) tensor words, Disc. Math. Lett. (2024) Vol. 13, 44-49. See p. 49.

Crossrefs

A004523(n+1) = T(n,2).

Programs

Haskell

a182210 n k = a182210_tabl !! (n-1) !! (k-1)
a182210_tabl = [[k*(n+1) `div` (k+1) | k <- [1..n]] | n <- [1..]]
-- Reinhard Zumkeller, Jul 08 2012

Maple

seq(seq(floor(k*(n+1)/(k+1)),k=1..n),n=1..15);

Mathematica

Flatten[Table[Floor[k*(n+1)/(k+1)],{n,0,20},{k,n}]] (* Harvey P. Dale, Jul 21 2015 *)

Formula

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