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A182210
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Triangle T(n,k) = floor(k*(n+1)/(k+1)), 1 <= k <= n.
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3
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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
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1,3
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COMMENTS
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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).
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LINKS
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FORMULA
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T(n,k) = floor(k(n+1)/(k+1)).
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EXAMPLE
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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,
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MAPLE
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seq(seq(floor(k*(n+1)/(k+1)), k=1..n), n=1..15);
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MATHEMATICA
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Flatten[Table[Floor[k*(n+1)/(k+1)], {n, 0, 20}, {k, n}]] (* Harvey P. Dale, Jul 21 2015 *)
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PROG
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(Haskell)
a182210 n k = a182210_tabl !! (n-1) !! (k-1)
a182210_tabl = [[k*(n+1) `div` (k+1) | k <- [1..n]] | n <- [1..]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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