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A182210
Triangle T(n,k) = floor(k*(n+1)/(k+1)), 1 <= k <= n.
3
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
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
KEYWORD
nonn,nice,easy,tabl,look
AUTHOR
Dennis P. Walsh, Apr 18 2012
STATUS
approved