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A238432 Number of compositions of n avoiding equidistant 3-term arithmetic progressions.

%I A238432 #18 May 21 2018 06:39:50
%S A238432 1,1,2,3,7,13,22,41,74,133,233,400,714,1209,2091,3591,6089,10316,
%T A238432 17477,29413,49515,82474,137659,228461,377936,623710,1025445,1680418,
%U A238432 2746242,4474654,7270430,11774128,19020802,30640812,49222427,78857338,126033488,200872080
%N A238432 Number of compositions of n avoiding equidistant 3-term arithmetic progressions.
%H A238432 Joerg Arndt and Alois P. Heinz, <a href="/A238432/b238432.txt">Table of n, a(n) for n = 0..45</a>
%e A238432 The a(5) = 13 such compositions are:
%e A238432 01:  [ 1 1 2 1 ]
%e A238432 02:  [ 1 1 3 ]
%e A238432 03:  [ 1 2 1 1 ]
%e A238432 04:  [ 1 2 2 ]
%e A238432 05:  [ 1 3 1 ]
%e A238432 06:  [ 1 4 ]
%e A238432 07:  [ 2 1 2 ]
%e A238432 08:  [ 2 2 1 ]
%e A238432 09:  [ 2 3 ]
%e A238432 10:  [ 3 1 1 ]
%e A238432 11:  [ 3 2 ]
%e A238432 12:  [ 4 1 ]
%e A238432 13:  [ 5 ]
%e A238432 Note that the first and third composition contain the progression 1,1,1, but not in equidistant positions.
%p A238432 b:= proc(n, l) local j;
%p A238432       for j from 2 to iquo(nops(l)+1, 2) do
%p A238432       if l[1]-l[j]=l[j]-l[2*j-1] then return 0 fi od;
%p A238432      `if`(n=0, 1, add(b(n-i, [i, l[]]), i=1..n))
%p A238432     end:
%p A238432 a:= n-> b(n, []):
%p A238432 seq(a(n), n=0..20);
%t A238432 b[n_, l_] := b[n, 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, Sum[b[n - i, Prepend[l, i]], {i, 1, n}]]];
%t A238432 a[n_] := b[n, {}];
%t A238432 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y A238432 Cf. A238433 (same for partitions).
%Y A238432 Cf. A238569 (compositions avoiding any 3-term arithmetic progression).
%Y A238432 Cf. A238423 (compositions avoiding three consecutive parts in arithmetic progression).
%Y A238432 Cf. A238686.
%K A238432 nonn
%O A238432 0,3
%A A238432 _Joerg Arndt_ and _Alois P. Heinz_, Mar 01 2014