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

%I A238433 #21 Jan 12 2021 09:06:52
%S A238433 1,1,2,2,4,5,6,8,13,12,19,23,29,35,45,52,68,80,98,111,141,163,198,230,
%T A238433 283,320,376,443,517,585,719,799,932,1085,1254,1417,1668,1861,2138,
%U A238433 2449,2804,3166,3666,4083,4662,5277,5960,6676,7651,8494,9635,10803,12157
%N A238433 Number of partitions of n avoiding equidistant 3-term arithmetic progressions.
%H A238433 Fausto A. C. Cariboni, <a href="/A238433/b238433.txt">Table of n, a(n) for n = 0..300</a> (terms 0..150 from Joerg Arndt and Alois P. Heinz)
%e A238433 The a(8) = 13 such partitions are:
%e A238433 01:   [ 1 1 2 4 ]
%e A238433 02:   [ 1 1 3 3 ]
%e A238433 03:   [ 1 1 6 ]
%e A238433 04:   [ 1 2 2 3 ]
%e A238433 05:   [ 1 2 5 ]
%e A238433 06:   [ 1 3 4 ]
%e A238433 07:   [ 1 7 ]
%e A238433 08:   [ 2 2 4 ]
%e A238433 09:   [ 2 3 3 ]
%e A238433 10:   [ 2 6 ]
%e A238433 11:   [ 3 5 ]
%e A238433 12:   [ 4 4 ]
%e A238433 13:   [ 8 ]
%e A238433 Note that the fourth partition has the arithmetic progression 1,2,3, but not in equidistant positions.
%p A238433 b:= proc(n, i, l) local j;
%p A238433       for j from 2 to iquo(nops(l)+1, 2) do
%p A238433       if l[1]-l[j]=l[j]-l[2*j-1] then return 0 fi od;
%p A238433      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, l)+
%p A238433      `if`(i>n, 0, b(n-i, i, [i,l[]]))))
%p A238433     end:
%p A238433 a:= n-> b(n, n, []):
%p A238433 seq(a(n), n=0..40);
%t A238433 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]]]]]];
%t A238433 a[n_] := b[n, n, {}];
%t A238433 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y A238433 Cf. A238432 (same for compositions).
%Y A238433 Cf. A238571 (partitions avoiding any 3-term arithmetic progression).
%Y A238433 Cf. A238424 (partitions avoiding three consecutive parts in arithmetic progression).
%Y A238433 Cf. A238687.
%K A238433 nonn
%O A238433 0,3
%A A238433 _Joerg Arndt_ and _Alois P. Heinz_, Mar 01 2014