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.
Original entry on oeis.org
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
Offset: 0
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].
Cf.
A238687 (the same for partitions).
-
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);
-
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, {}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)
A238433
Number of partitions of n avoiding equidistant 3-term arithmetic progressions.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 6, 8, 13, 12, 19, 23, 29, 35, 45, 52, 68, 80, 98, 111, 141, 163, 198, 230, 283, 320, 376, 443, 517, 585, 719, 799, 932, 1085, 1254, 1417, 1668, 1861, 2138, 2449, 2804, 3166, 3666, 4083, 4662, 5277, 5960, 6676, 7651, 8494, 9635, 10803, 12157
Offset: 0
The a(8) = 13 such partitions are:
01: [ 1 1 2 4 ]
02: [ 1 1 3 3 ]
03: [ 1 1 6 ]
04: [ 1 2 2 3 ]
05: [ 1 2 5 ]
06: [ 1 3 4 ]
07: [ 1 7 ]
08: [ 2 2 4 ]
09: [ 2 3 3 ]
10: [ 2 6 ]
11: [ 3 5 ]
12: [ 4 4 ]
13: [ 8 ]
Note that the fourth partition has the arithmetic progression 1,2,3, but not in equidistant positions.
Cf.
A238432 (same for compositions).
Cf.
A238571 (partitions avoiding any 3-term arithmetic progression).
Cf.
A238424 (partitions avoiding three consecutive parts in arithmetic progression).
-
b:= proc(n, i, l) local j;
for j from 2 to iquo(nops(l)+1, 2) do
if l[1]-l[j]=l[j]-l[2*j-1] then return 0 fi od;
`if`(n=0, 1, `if`(i<1, 0, b(n, i-1, l)+
`if`(i>n, 0, b(n-i, i, [i,l[]]))))
end:
a:= n-> b(n, n, []):
seq(a(n), n=0..40);
-
b[n_, i_, l_] := b[n, i, l] = Module[{j}, For[ j = 2 , j <= Quotient[ Length[l] + 1, 2] , j++, If[ l[[1]] - l[[j]] == l[[j]] - l[[2*j - 1]] , Return[0]]]; If[n == 0, 1, If[i < 1, 0, b[n, i - 1, l] + If[i > n, 0, b[n - i, i, Prepend[l, i]]]]]];
a[n_] := b[n, n, {}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 21 2018, translated from Maple *)
Showing 1-2 of 2 results.