A357643 - OEIS

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

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OFFSET

0,3

LINKS

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

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

EXAMPLE

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)

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}]

PROG

(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

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.

Sequence in context: A107237 A070047 A101198 * A034394 A058689 A173510

Adjacent sequences: A357640 A357641 A357642 * A357644 A357645 A357646

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 12 2022

EXTENSIONS

More terms from Alois P. Heinz, Oct 12 2022

STATUS

approved