A329738 Number of compositions of n whose run-lengths are all equal.

1, 1, 2, 4, 6, 8, 19, 24, 45, 75, 133, 215, 401, 662, 1177, 2035, 3587, 6190, 10933, 18979, 33339, 58157, 101958, 178046, 312088, 545478, 955321, 1670994, 2925717, 5118560, 8960946, 15680074, 27447350, 48033502, 84076143, 147142496, 257546243, 450748484, 788937192

Offset

0,3

Comments

A composition of n is a finite sequence of positive integers with sum n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{d|n} A003242(d).

a(n) = A329745(n) + A000005(n).

Example

The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (131)    (51)
                            (212)    (123)
                            (11111)  (132)
                                     (141)
                                     (213)
                                     (222)
                                     (231)
                                     (312)
                                     (321)
                                     (1122)
                                     (1212)
                                     (2121)
                                     (2211)
                                     (111111)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], SameQ@@Length/@Split[#]&]], {n, 0, 10}]

Prog

(PARI) seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); concat([1], vector(n, k, sumdiv(k, d, b[d])))} \\ Andrew Howroyd, Dec 30 2020

Crossrefs

Compositions with relatively prime run-lengths are A000740.

Compositions with equal multiplicities are A098504.

Compositions with equal differences are A175342.

Compositions with distinct run-lengths are A329739.

Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329745, A329766.

Sequence in context: A048784 A152491 A355557 * A133296 A323333 A076402

Adjacent sequences: A329735 A329736 A329737 * A329739 A329740 A329741

Keyword

nonn

Author

Gus Wiseman, Nov 20 2019

Status

approved