A342527 Number of compositions of n with alternating parts equal.

1, 1, 2, 4, 6, 8, 11, 12, 16, 17, 21, 20, 29, 24, 31, 32, 38, 32, 46, 36, 51, 46, 51, 44, 69, 51, 61, 60, 73, 56, 87, 60, 84, 74, 81, 76, 110, 72, 91, 88, 115, 80, 123, 84, 117, 112, 111, 92, 153, 101, 132, 116, 139, 104, 159, 120, 161, 130, 141, 116, 205, 120, 151, 156, 178, 142, 195, 132, 183, 158

Offset: 0

Author

Gus Wiseman, Mar 24 2021

Keywords

• nonn

Comments

These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

Examples

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (141)     (61)       (62)
                            (11111)  (222)     (151)      (71)
                                     (1212)    (232)      (161)
                                     (2121)    (313)      (242)
                                     (111111)  (12121)    (323)
                                               (1111111)  (1313)
                                                          (2222)
                                                          (3131)
                                                          (21212)
                                                          (11111111)
		

Links

• Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Crossrefs

The odd-length case is A062968.

The even-length case is A065608.

The version with alternating parts unequal is A224958 (unordered: A000726).

The version with alternating parts weakly decreasing is A342528.

A000005 counts constant compositions.

A000041 counts weakly increasing (or weakly decreasing) compositions.

A000203 adds up divisors.

A002843 counts compositions with all adjacent parts x <= 2y.

A003242 counts anti-run compositions.

A175342 counts compositions with constant differences.

A342495 counts compositions with constant first quotients.

A342496 counts partitions with constant first quotients (strict: A342515, ranking: A342522).

Cf. A001522, A008965, A048004, A059966, A064410, A064428, A069916, A114921, A167606, A325545, A325557.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

Formula

a(n) = 1 + n + A000203(n) - 2*A000005(n).

a(n) = A065608(n) + A062968(n).