%I #29 Mar 21 2022 10:07:00

%S 1,1,2,2,4,5,6,8,13,13,19,24,30,36,47,54,72,85,106,123,151,178,220,

%T 256,314,362,432,505,605,692,827,953,1121,1303,1522,1729,2037,2321,

%U 2691,3095,3577,4061,4699,5334,6126,6959,7966,9005,10317,11638,13252,14977

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

%C 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 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 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>

%e The a(8) = 13 such partitions are:

%e 01: [ 3 2 2 1 ]

%e 02: [ 3 3 1 1 ]

%e 03: [ 3 3 2 ]

%e 04: [ 4 2 1 1 ]

%e 05: [ 4 2 2 ]

%e 06: [ 4 3 1 ]

%e 07: [ 4 4 ]

%e 08: [ 5 2 1 ]

%e 09: [ 5 3 ]

%e 10: [ 6 1 1 ]

%e 11: [ 6 2 ]

%e 12: [ 7 1 ]

%e 13: [ 8 ]

%t 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 Table[Length[Select[IntegerPartitions[n],!MemberQ[Differences[#,2],0]&]],{n,0,30}] (* _Gus Wiseman_, Mar 31 2020 *)

%Y Cf. A238433 (partitions avoiding equidistant arithmetic progressions).

%Y Cf. A238571 (partitions avoiding any arithmetic progression).

%Y Cf. A238687.

%Y The version for compositions is A238423, with strict case A325849.

%Y The version for permutations is A295370.

%Y The strict case is A332668.

%Y The Heinz numbers of these partitions are the complement of A333195.

%Y Partitions with equal differences are A049988.

%Y Cf. A007862, A175342, A325850, A325851, A325852, A325874, A333631.

%K nonn

%O 0,3

%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 26 2014