

A182210


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


1



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
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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


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 12game 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 !! (n1) !! (k1)
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: A342739 A137734 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



