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A131044
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Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n into k parts such that at least two adjacent parts are equal.
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15
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1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 4, 4, 1, 1, 1, 3, 8, 5, 1, 1, 0, 6, 14, 14, 6, 1, 1, 1, 6, 21, 32, 21, 7, 1, 1, 0, 7, 32, 55, 54, 28, 8, 1, 1, 1, 8, 38, 96, 116, 83, 36, 9, 1, 1, 0, 10, 54, 142, 222, 206, 120, 45, 10, 1, 1, 1, 9, 65, 211, 386, 438, 328, 165, 55, 11, 1
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OFFSET
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1,9
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COMMENTS
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Condition is void for compositions into 1 part (there is one such composition).
Triangle = Pascal's triangle (A007318) - A106351, except for first column.
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LINKS
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EXAMPLE
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T(5,3) = 4 because among the 6 compositions of 5 into 3 parts there are 4 with one part repeated, marked by (*) between the parts:
[ 3 1*1 ], [ 2*2 1 ], [ 1 3 1 ], [ 2 1 2 ], [ 1 2*2 ], [ 1*1 3 ].
Triangle begins:
1;
1, 1;
1, 0, 1;
1, 1, 2, 1;
1, 0, 4, 4, 1;
1, 1, 3, 8, 5, 1;
1, 0, 6, 14, 14, 6, 1;
1, 1, 6, 21, 32, 21, 7, 1;
...
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MAPLE
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b:= proc(n, h, t) option remember;
if n<t then 0 elif n=0 then `if`(t=0, 1, 0)
else add(`if`(h=j, 0, b(n-j, j, t-1)), j=1..n) fi
end:
T:= (n, k)-> `if`(k=1, 1, binomial(n-1, k-1) -b(n, -1, k)):
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MATHEMATICA
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b[n_, h_, t_] := b[n, h, t] = Which[n<t, 0, n == 0, If[t == 0, 1, 0], True, Sum[If[h == j, 0, b[n-j, j, t-1]], {j, 1, n}]]; T[n_, k_] := If[k == 1, 1, Binomial[n-1, k-1] - b[n, -1, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
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PROG
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(Sage)
allowed = lambda x: len(x) <= 1 or 0 in differences(x)
return len([c for c in Compositions(n, length=k) if allowed(c)])
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CROSSREFS
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Cf. A106351 (no two adjacent parts are equal).
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KEYWORD
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AUTHOR
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STATUS
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approved
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