A332668 Number of strict integer partitions of n without three consecutive parts in arithmetic progression.

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 20, 19, 26, 31, 34, 41, 50, 53, 67, 78, 84, 99, 120, 130, 154, 177, 193, 226, 262, 291, 332, 375, 419, 479, 543, 608, 676, 765, 859, 961, 1075, 1202, 1336, 1495, 1672, 1854, 2050, 2301, 2536, 2814, 3142, 3448, 3809

Offset

0,4

Comments

Also the number of strict integer partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..450

Wikipedia, Arithmetic progression

Example

The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)
                        (41)  (51)  (52)   (62)   (63)   (73)
                                    (61)   (71)   (72)   (82)
                                    (421)  (431)  (81)   (91)
                                           (521)  (621)  (532)
                                                         (541)
                                                         (631)
                                                         (721)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!MatchQ[Differences[#], {___, x_, x_, ___}]&]], {n, 0, 30}]

Crossrefs

Anti-run compositions are counted by A003242.

Normal anti-runs of length n + 1 are counted by A005649.

Strict partitions with equal differences are A049980.

Partitions with equal differences are A049988.

The non-strict version is A238424.

The version for permutations is A295370.

Anti-run compositions are ranked by A333489.

Cf. A006560, A007862, A238423, A307824, A325328, A325852, A325874, A333195.

Sequence in context: A320347 A178932 A325852 * A206439 A097450 A062303

Adjacent sequences: A332665 A332666 A332667 * A332669 A332670 A332671

Keyword

nonn

Author

Gus Wiseman, Mar 28 2020

Status

approved