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A238424 Number of partitions of n without three consecutive parts in arithmetic progression.

%I A238424 #29 Mar 21 2022 10:07:00
%S A238424 1,1,2,2,4,5,6,8,13,13,19,24,30,36,47,54,72,85,106,123,151,178,220,
%T A238424 256,314,362,432,505,605,692,827,953,1121,1303,1522,1729,2037,2321,
%U A238424 2691,3095,3577,4061,4699,5334,6126,6959,7966,9005,10317,11638,13252,14977
%N A238424 Number of partitions of n without three consecutive parts in arithmetic progression.
%C A238424 Also the number of partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences. - _Gus Wiseman_, Mar 31 2020
%H A238424 Vaclav Kotesovec, <a href="/A238424/b238424.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Joerg Arndt and Alois P. Heinz, terms 301..350 from Fausto A. C. Cariboni)
%H A238424 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>
%e A238424 The a(8) = 13 such partitions are:
%e A238424 01:  [ 3 2 2 1 ]
%e A238424 02:  [ 3 3 1 1 ]
%e A238424 03:  [ 3 3 2 ]
%e A238424 04:  [ 4 2 1 1 ]
%e A238424 05:  [ 4 2 2 ]
%e A238424 06:  [ 4 3 1 ]
%e A238424 07:  [ 4 4 ]
%e A238424 08:  [ 5 2 1 ]
%e A238424 09:  [ 5 3 ]
%e A238424 10:  [ 6 1 1 ]
%e A238424 11:  [ 6 2 ]
%e A238424 12:  [ 7 1 ]
%e A238424 13:  [ 8 ]
%t A238424 a[n_,r_,d_] := a[n,r,d] = Block[{j}, If[n == 0, 1, Sum[If[j == r+d, 0, a[n-j, j, j - r]], {j, Min[n, r]}]]]; a[n_] := a[n, 2*n+1, 0]; a /@ Range[0, 100] (* _Giovanni Resta_, Mar 02 2014 *)
%t A238424 Table[Length[Select[IntegerPartitions[n],!MemberQ[Differences[#,2],0]&]],{n,0,30}] (* _Gus Wiseman_, Mar 31 2020 *)
%Y A238424 Cf. A238433 (partitions avoiding equidistant arithmetic progressions).
%Y A238424 Cf. A238571 (partitions avoiding any arithmetic progression).
%Y A238424 Cf. A238687.
%Y A238424 The version for compositions is A238423, with strict case A325849.
%Y A238424 The version for permutations is A295370.
%Y A238424 The strict case is A332668.
%Y A238424 The Heinz numbers of these partitions are the complement of A333195.
%Y A238424 Partitions with equal differences are A049988.
%Y A238424 Cf. A007862, A175342, A325850, A325851, A325852, A325874, A333631.
%K A238424 nonn
%O A238424 0,3
%A A238424 _Joerg Arndt_ and _Alois P. Heinz_, Feb 26 2014