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

0,3

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

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

Example

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 ]

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 *)

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.

Sequence in context: A293957 A238687 A238433 * A121269 A211860 A250114

Adjacent sequences: A238421 A238422 A238423 * A238425 A238426 A238427

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Status

approved