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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

%I

%S 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,

%T 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,

%U 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

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

%D For references see A027375.

%H B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="http://arxiv.org/abs/1212.6102">On Curling Numbers of Integer Sequences</a>, arXiv:1212.6102 [math.CO], Dec 25 2012.

%H B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Sloane/sloane3.html">On Curling Numbers of Integer Sequences</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.

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

%e Triangle begins:

%e 2,

%e 2, 2,

%e 2, 0, 6,

%e 2, 2, 0, 12,

%e 2, 0, 0, 0, 30,

%e 2, 2, 6, 0, 0, 54,

%e 2, 0, 0, 0, 0, 0, 126,

%e 2, 2, 0, 12, 0, 0, 0, 240,

%e 2, 0, 6, 0, 0, 0, 0, 0, 504,

%e 2, 2, 0, 0, 30, 0, 0, 0, 0, 990,

%e ...

%e For n=4 the 16 sequences are:

%e 0000, 1111, period 1,

%e 0101, 1010, period 2,

%e and the rest have period 4.

%p with(numtheory): A027375:=n->add( mobius(d)*2^(n/d), d in divisors(n));

%p a:=proc(n,k) global A027375;

%p if n mod k = 0 then A027375(k) else 0; fi; end;

%t a027375[n_] := DivisorSum[n, MoebiusMu[n/#]*2^#&];

%t T[n_, k_] := If[Divisible[n, k], a027375[k], 0];

%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-Fran├žois Alcover_, Nov 26 2017 *)

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

%K nonn,tabl

%O 1,1

%A _N. J. A. Sloane_, Sep 25 2012

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Last modified September 18 15:48 EDT 2020. Contains 337169 sequences. (Running on oeis4.)