A238433 Number of partitions of n avoiding equidistant 3-term arithmetic progressions.

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

Author

Joerg Arndt and Alois P. Heinz, Mar 01 2014

Keywords

• nonn

Examples

			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.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300 (terms 0..150 from Joerg Arndt and Alois P. Heinz)

Crossrefs

Cf. A238432 (same for compositions).

Cf. A238571 (partitions avoiding any 3-term arithmetic progression).

Cf. A238424 (partitions avoiding three consecutive parts in arithmetic progression).

Cf. A238687.

Programs

Maple

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);

Mathematica

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 *)