A325849 Number of strict compositions of n with no three consecutive parts in arithmetic progression.

1, 1, 1, 3, 3, 5, 9, 13, 19, 23, 51, 57, 91, 117, 179, 283, 381, 531, 737, 1017, 1335, 2259, 2745, 3983, 5289, 7367, 9413, 13155, 19461, 25129, 33997, 45633, 61225, 80481, 107091, 137475, 205243, 253997, 345527, 447003, 604919, 768331, 1026167, 1299227

Offset: 0

Author

Gus Wiseman, May 31 2019

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers with sum n. a(n) is the number of strict compositions of n with no two of their adjacent first-differences equal, or with no 0's in their second-differences.

Examples

			The a(1) = 1 through a(8) = 19 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)
            (12)  (13)  (14)  (15)   (16)   (17)
            (21)  (31)  (23)  (24)   (25)   (26)
                        (32)  (42)   (34)   (35)
                        (41)  (51)   (43)   (53)
                              (132)  (52)   (62)
                              (213)  (61)   (71)
                              (231)  (124)  (125)
                              (312)  (142)  (134)
                                     (214)  (143)
                                     (241)  (152)
                                     (412)  (215)
                                     (421)  (251)
                                            (314)
                                            (341)
                                            (413)
                                            (431)
                                            (512)
                                            (521)
		

Links

• Wikipedia, Arithmetic progression

Crossrefs

The non-strict case is A238423.

Cf. A007862, A049988, A175342, A279945, A295370, A325545, A325874, A325875.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],!MemberQ[Differences[#,2],0]&]],{n,0,30}]