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A238686
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Number of compositions c of n such that no three points (i,c_i), (j,c_j), (k,c_k) are collinear, where c_i denotes the i-th part.
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6
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1, 1, 2, 3, 7, 11, 19, 30, 53, 87, 148, 219, 365, 555, 884, 1379, 2098, 3152, 4865, 7051, 10884, 15681, 23637, 34062, 50336, 72425, 105738, 149781, 217625, 308859, 440889, 623823, 885116, 1241075, 1744784, 2433371, 3401728, 4719635, 6548306, 9035003, 12472106
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OFFSET
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0,3
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LINKS
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EXAMPLE
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There are a(6) = 19 such compositions of 6: [6], [5,1], [4,2], [3,3], [2,4], [1,5], [4,1,1], [2,3,1], [1,4,1], [1,3,2], [3,1,2], [2,1,3], [1,1,4], [2,2,1,1], [1,2,2,1], [2,1,2,1], [1,2,1,2], [2,1,1,2], [1,1,2,2].
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MAPLE
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b:= proc(n, l) local j, k, m; m:= nops(l);
for j to m-2 do for k from j+1 to m-1 do
if (l[m]-l[k])*(k-j)=(l[k]-l[j])*(m-k)
then return 0 fi od od;
`if`(n=0, 1, add(b(n-i, [l[], i]), i=1..n))
end:
a:= n-> b(n, []):
seq(a(n), n=0..20);
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MATHEMATICA
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b[n_, l_] := Module[{j, k, m = Length[l]}, For[ j = 1, j <= m - 2, j++, For[k = j+1, k <= m - 1 , k++, If[(l[[m]] - l[[k]])*(k - j) == (l[[k]] - l[[j]])*(m - k), Return[0]]]]; If[n == 0, 1, Sum[b[n - i, Append[l, i]], {i, 1, n}]]];
a[n_] := b[n, {}];
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CROSSREFS
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Cf. A238687 (the same for partitions).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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