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

0,3

Links

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

Example

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.

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 *)

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.

Sequence in context: A325874 A293957 A238687 * A238424 A121269 A211860

Adjacent sequences: A238430 A238431 A238432 * A238434 A238435 A238436

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Mar 01 2014

Status

approved