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