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A238686 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.

%I A238686 #27 Mar 23 2022 04:29:20
%S A238686 1,1,2,3,7,11,19,30,53,87,148,219,365,555,884,1379,2098,3152,4865,
%T A238686 7051,10884,15681,23637,34062,50336,72425,105738,149781,217625,308859,
%U A238686 440889,623823,885116,1241075,1744784,2433371,3401728,4719635,6548306,9035003,12472106
%N A238686 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.
%H A238686 Fausto A. C. Cariboni, <a href="/A238686/b238686.txt">Table of n, a(n) for n = 0..74</a> (terms 0..50 from Joerg Arndt and Alois P. Heinz).
%e A238686 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].
%p A238686 b:= proc(n, l) local j, k, m; m:= nops(l);
%p A238686       for j to m-2 do for k from j+1 to m-1 do
%p A238686         if (l[m]-l[k])*(k-j)=(l[k]-l[j])*(m-k)
%p A238686           then return 0 fi od od;
%p A238686      `if`(n=0, 1, add(b(n-i, [l[], i]), i=1..n))
%p A238686     end:
%p A238686 a:= n-> b(n, []):
%p A238686 seq(a(n), n=0..20);
%t A238686 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}]]];
%t A238686 a[n_] := b[n, {}];
%t A238686 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y A238686 Cf. A238687 (the same for partitions).
%Y A238686 Cf. A238423, A238432, A238569.
%K A238686 nonn
%O A238686 0,3
%A A238686 _Joerg Arndt_ and _Alois P. Heinz_, Mar 02 2014