A216953 Triangle read by rows: T(n,k) (n>=1, 1<=k<=n) = number of binary sequences of length n with minimal period k.

2, 2, 2, 2, 0, 6, 2, 2, 0, 12, 2, 0, 0, 0, 30, 2, 2, 6, 0, 0, 54, 2, 0, 0, 0, 0, 0, 126, 2, 2, 0, 12, 0, 0, 0, 240, 2, 0, 6, 0, 0, 0, 0, 0, 504, 2, 2, 0, 0, 30, 0, 0, 0, 0, 990, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2046, 2, 2, 6, 12, 0, 54, 0, 0, 0, 0, 0, 4020, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8190, 2, 2, 0, 0, 0, 0, 126, 0, 0, 0, 0, 0, 0, 16254

Offset

1,1

References

For references see A027375.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..20100

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102 [math.CO], 2012-2013.

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.

Formula

If k divides n, T(n,k) = A027375(k), otherwise 0.

Example

Triangle begins:
  2,
  2, 2,
  2, 0, 6,
  2, 2, 0, 12,
  2, 0, 0, 0, 30,
  2, 2, 6, 0, 0, 54,
  2, 0, 0, 0, 0, 0, 126,
  2, 2, 0, 12, 0, 0, 0, 240,
  2, 0, 6, 0, 0, 0, 0, 0, 504,
  2, 2, 0, 0, 30, 0, 0, 0, 0, 990,
  ...
For n=4 the 16 sequences are:
  0000, 1111, period 1,
  0101, 1010, period 2,
and the rest have period 4.

Maple

with(numtheory): A027375:=n->add( mobius(d)*2^(n/d), d in divisors(n));
a:=proc(n, k) global A027375;
if n mod k = 0 then A027375(k) else 0; fi; end;

Mathematica

a027375[n_] := DivisorSum[n, MoebiusMu[n/#]*2^#&];
T[n_, k_] := If[Divisible[n, k], a027375[k], 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 26 2017 *)

Prog

(Magma) a027375 := func<n | &+[MoebiusMu(n div d)*2^d : d in Divisors(n)]>; T := func<n, k | (n mod k eq 0) select a027375(k) else 0>; seq := []; for n in [1..14] do for k in [1..n] do Append(~seq, T(n, k)); end for; end for; seq; // Vincenzo Librandi, Mar 16 2026

Crossrefs

Cf. A027375 (the main diagonal), A216954, A001037.

Sequence in context: A268242 A362932 A309509 * A326786 A276206 A334222

Adjacent sequences: A216950 A216951 A216952 * A216954 A216955 A216956

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 25 2012

Status

approved