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

1, 1, 2, 2, 4, 5, 6, 8, 13, 13, 19, 24, 30, 36, 47, 54, 72, 85, 106, 123, 151, 178, 220, 256, 314, 362, 432, 505, 605, 692, 827, 953, 1121, 1303, 1522, 1729, 2037, 2321, 2691, 3095, 3577, 4061, 4699, 5334, 6126, 6959, 7966, 9005, 10317, 11638, 13252, 14977

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Keywords

• nonn

Comments

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

Examples

			The a(8) = 13 such partitions are:
01:  [ 3 2 2 1 ]
02:  [ 3 3 1 1 ]
03:  [ 3 3 2 ]
04:  [ 4 2 1 1 ]
05:  [ 4 2 2 ]
06:  [ 4 3 1 ]
07:  [ 4 4 ]
08:  [ 5 2 1 ]
09:  [ 5 3 ]
10:  [ 6 1 1 ]
11:  [ 6 2 ]
12:  [ 7 1 ]
13:  [ 8 ]
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..300 from Joerg Arndt and Alois P. Heinz, terms 301..350 from Fausto A. C. Cariboni)
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

Crossrefs

Cf. A238433 (partitions avoiding equidistant arithmetic progressions).

Cf. A238571 (partitions avoiding any arithmetic progression).

Cf. A238687.

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

The version for permutations is A295370.

The strict case is A332668.

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

Partitions with equal differences are A049988.

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

Programs

Mathematica

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