A357643 Number of integer compositions of n into parts that are alternately equal and unequal.

1, 1, 2, 1, 3, 3, 5, 5, 9, 7, 17, 14, 28, 25, 49, 42, 87, 75, 150, 132, 266, 226, 466, 399, 810, 704, 1421, 1223, 2488, 2143, 4352, 3759, 7621, 6564, 13339, 11495, 23339, 20135, 40852, 35215, 71512, 61639, 125148, 107912, 219040, 188839, 383391, 330515, 670998

Offset: 0

Author

Gus Wiseman, Oct 12 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)  (4)    (5)    (6)     (7)      (8)
       (11)       (22)   (113)  (33)    (115)    (44)
                  (112)  (221)  (114)   (223)    (116)
                                (1122)  (331)    (224)
                                (2211)  (11221)  (332)
                                                 (1133)
                                                 (3311)
                                                 (22112)
                                                 (112211)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

The even-length version is A003242, ranked by A351010, partitions A035457.

Without equal relations we have A016116, equal only A001590 (apparently).

The version for partitions is A351005.

The opposite version is A357644, partitions A351006.

A011782 counts compositions.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357645 counts compositions by half-alternating sum, skew A357646.

Cf. A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,15}]

PARI

C_x(N) = {my(x='x+O('x^N), h=(1+sum(k=1,N, (x^k)/(1+x^(2*k))))/(1-sum(k=1,N, (x^(2*k))/(1+x^(2*k))))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, May 28 2024

Formula

G.f.: (1 + Sum_{k>0} (x^k)/(1 + x^(2*k)))/(1 - Sum_{k>0} (x^(2*k))/(1 + x^(2*k))). - John Tyler Rascoe, May 28 2024

Extensions

More terms from Alois P. Heinz, Oct 12 2022