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

Author

Gus Wiseman, Mar 28 2020

Keywords

• nonn

Comments

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

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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