A296975 - OEIS

%I #12 Aug 29 2018 02:52:55

%S 1,2,12,72,540,4668,47292,545760,7087248,102247020,1622632572,

%T 28091562840,526858348380,10641342923148,230283190977300,

%U 5315654681435520,130370767029135900,3385534663249753392,92801587319328411132,2677687796244281955480,81124824998504073834516

%N Number of aperiodic normal sequences of length n.

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

%H Andrew Howroyd, <a href="/A296975/b296975.txt">Table of n, a(n) for n = 1..200</a>

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

%e The a(3) = 12 aperiodic normal sequences are 112, 121, 122, 123, 132, 211, 212, 213, 221, 231, 312, 321.

%e 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.

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

%o (PARI) \\ here b(n) is A000670.

%o b(n)={polcoef(serlaplace(1/(2-exp(x+O(x*x^n)))),n)}

%o a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ _Andrew Howroyd_, Aug 29 2018

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 22 2017