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

1,3

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).

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.

Formula

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

Example

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,

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A137734 A366066 A352627 * A078705 A346018 A050331

Adjacent sequences: A182207 A182208 A182209 * A182211 A182212 A182213

Keyword

nonn,nice,easy,tabl,look

Author

Dennis P. Walsh, Apr 18 2012

Status

approved