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A238687 Number of partitions p of n such that no three points (i,p_i), (j,p_j), (k,p_k) are collinear, where p_i denotes the i-th part.

%I A238687 #20 Feb 09 2021 02:05:31
%S A238687 1,1,2,2,4,5,6,8,13,10,18,21,27,29,41,41,62,65,77,91,114,127,151,173,
%T A238687 213,232,279,322,372,410,491,518,630,724,814,894,1057,1141,1326,1502,
%U A238687 1681,1839,2146,2324,2636,2966,3272,3607,4173,4422,5035,5616,6195,6703
%N A238687 Number of partitions p of n such that no three points (i,p_i), (j,p_j), (k,p_k) are collinear, where p_i denotes the i-th part.
%H A238687 Fausto A. C. Cariboni, <a href="/A238687/b238687.txt">Table of n, a(n) for n = 0..300</a> (terms 0..150 from Alois P. Heinz)
%e A238687 There are a(10) = 18 such partitions of 10: [6,2,1,1], [5,2,2,1], [4,4,1,1], [3,3,2,2], [8,1,1], [7,2,1], [6,3,1], [6,2,2], [5,4,1], [5,3,2], [4,4,2], [4,3,3], [9,1], [8,2], [7,3], [6,4], [5,5], [10].
%p A238687 b:= proc(n, i, l) local j, k, m; m:= nops(l);
%p A238687       for j to m-2 do for k from j+1 to m-1 do
%p A238687         if (l[m]-l[k])*(k-j)=(l[k]-l[j])*(m-k)
%p A238687           then return 0 fi od od;
%p A238687      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, l)+
%p A238687      `if`(i>n, 0, b(n-i, i, [l[], i]))))
%p A238687     end:
%p A238687 a:= n-> b(n, n, []):
%p A238687 seq(a(n), n=0..40);
%t A238687 b[n_, i_, 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, If[i < 1, 0, b[n, i - 1, l] + If[i > n, 0, b[n - i, i, Append[l, i]]]]]];
%t A238687 a[n_] := b[n, n, {}];
%t A238687 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y A238687 Cf. A238686 (the same for compositions).
%Y A238687 Cf. A238424, A238433, A238571.
%K A238687 nonn
%O A238687 0,3
%A A238687 _Joerg Arndt_ and _Alois P. Heinz_, Mar 02 2014