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

Author

N. J. A. Sloane, Sep 25 2012

Keywords

• nonn
• tabl

Examples

			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.
		

References

• For references see A027375.

Links

• B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102 [math.CO], Dec 25 2012.
• 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.

Crossrefs

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

Programs

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 *)

Formula

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