A296975 Number of aperiodic normal sequences of length n.

1, 2, 12, 72, 540, 4668, 47292, 545760, 7087248, 102247020, 1622632572, 28091562840, 526858348380, 10641342923148, 230283190977300, 5315654681435520, 130370767029135900, 3385534663249753392, 92801587319328411132, 2677687796244281955480, 81124824998504073834516

Offset

1,2

Comments

A finite sequence is normal if it spans an initial interval of positive integers. It is aperiodic if every cyclic rotation is different.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

a(n) = n * A060223(n) = Sum_{d|n} mu(d) * A000670(n/d).

Example

The a(3) = 12 aperiodic normal sequences are 112, 121, 122, 123, 132, 211, 212, 213, 221, 231, 312, 321.
The 15 non-aperiodic normal sequences of length 6 are: 111111, 112112, 121121, 121212, 122122, 123123, 132132, 211211, 212121, 212212, 213213, 221221, 231231, 312312, 321321.

Mathematica

Table[DivisorSum[n, MoebiusMu[n/#]*Sum[k!*StirlingS2[#, k], {k, #}]&], {n, 25}]

Prog

(PARI) \\ here b(n) is A000670.
b(n)={polcoef(serlaplace(1/(2-exp(x+O(x*x^n)))), n)}
a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ Andrew Howroyd, Aug 29 2018

Crossrefs

Cf. A000670, A000740, A001037, A019536, A027375, A060223, A095684, A185700, A296976, A296977, A296978.

Sequence in context: A371039 A052833 A277490 * A144086 A358064 A348767

Adjacent sequences: A296972 A296973 A296974 * A296976 A296977 A296978

Keyword

nonn

Author

Gus Wiseman, Dec 22 2017

Status

approved