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A274836 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of three.

%I A274836 #10 May 15 2018 08:34:16
%S A274836 1,1,1,1,2,3,4,9,18,33,89,215,481,1486,4187,10974,37907,121114,362953,
%T A274836 1385575,4924557,16494655,68685792,268113185,990074770,4455129525,
%U A274836 18896355932,76116156873,367577989487,1679905933299,7313571105815,37669220146964
%N A274836 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of three.
%H A274836 Alois P. Heinz, <a href="/A274836/b274836.txt">Table of n, a(n) for n = 0..721</a>
%H A274836 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A274836 a(5) = 3: 14|25|3, 1|25|3|4, 1|2|3|4|5.
%e A274836 a(6) = 4: 14|25|36, 1|25|36|4, 1|2|36|4|5, 1|2|3|4|5|6.
%e A274836 a(7) = 9: 147|25|36, 14|25|36|7, 17|25|36|4, 1|25|36|47, 17|2|36|4|5, 1|2|36|47|5, 17|2|3|4|5|6, 1|2|3|47|5|6, 1|2|3|4|5|6|7.
%p A274836 b:= proc(n, m, t) option remember; `if`(n=0, 1,
%p A274836      add(`if`(irem(j-t, 3)=0, b(n-1, max(m, j),
%p A274836               irem(t+1, 3)), 0), j=1..m+1))
%p A274836     end:
%p A274836 a:= n-> b(n, 0, 1):
%p A274836 seq(a(n), n=0..35);
%t A274836 b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j-t, 3] == 0, b[n-1, Max[m, j], Mod[t+1, 3]], 0], {j, 1, m+1}]];
%t A274836 a[n_] := b[n, 0, 1];
%t A274836 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 15 2018, after _Alois P. Heinz_ *)
%Y A274836 Column k=3 of A274835.
%K A274836 nonn
%O A274836 0,5
%A A274836 _Alois P. Heinz_, Jul 08 2016