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A261981
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Number T(n,k) of compositions of n such that k is the minimal distance between two identical parts; triangle T(n,k), n>=2, 1<=k<=floor((sqrt(8*n-7)-1)/2), read by rows.
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6
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1, 1, 4, 1, 9, 2, 18, 3, 41, 8, 2, 89, 16, 4, 185, 34, 10, 388, 57, 10, 810, 113, 30, 6, 1670, 213, 52, 12, 3435, 396, 104, 28, 7040, 733, 176, 50, 14360, 1333, 278, 62, 29226, 2419, 512, 152, 24, 59347, 4400, 878, 246, 48, 120229, 7934, 1492, 458, 108
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OFFSET
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2,3
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LINKS
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FORMULA
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T((n+1)*(n+2)/2+1,n+1) = A000142(n) for n>=0.
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EXAMPLE
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T(5,1) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.
T(5,2) = 2: 131, 212.
T(7,2) = 8: 151, 313, 232, 3121, 1213, 2131, 1312, 12121.
T(7,3) = 2: 1231, 1321.
Triangle T(n,k) begins:
n\k: 1 2 3 4 5
---+---------------------------
02 : 1;
03 : 1;
04 : 4, 1;
05 : 9, 2;
06 : 18, 3;
07 : 41, 8, 2;
08 : 89, 16, 4;
09 : 185, 34, 10;
10 : 388, 57, 10;
11 : 810, 113, 30, 6;
12 : 1670, 213, 52, 12;
13 : 3435, 396, 104, 28;
14 : 7040, 733, 176, 50;
15 : 14360, 1333, 278, 62;
16 : 29226, 2419, 512, 152, 24;
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MAPLE
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b:= proc(n, l) option remember;
`if`(n=0, 1, add(`if`(j in l, 0, b(n-j,
`if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
end:
T:= (n, k)-> b(n, [0$(k-1)])-b(n, [0$k]):
seq(seq(T(n, k), k=1..floor((sqrt(8*n-7)-1)/2)), n=2..20);
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MATHEMATICA
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b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]];
A[n_, k_] := b[n, Array[0&, Min[n, k]]];
T[n_, k_] := A[n, k-1] - A[n, k];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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