A323862 Table read by antidiagonals where A(n,k) is the number of n X k binary arrays in which both the sequence of rows and the sequence of columns are (independently) aperiodic.

2, 2, 2, 6, 10, 6, 12, 54, 54, 12, 30, 228, 498, 228, 30, 54, 990, 4020, 4020, 990, 54, 126, 3966, 32730, 65040, 32730, 3966, 126, 240, 16254, 261522, 1047540, 1047540, 261522, 16254, 240, 504, 65040, 2097018, 16768860, 33554370, 16768860, 2097018, 65040, 504

Offset

1,1

Comments

A sequence of length n is aperiodic if all n rotations of its entries are distinct.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Formula

A(n,k) = Sum_{d|n, e|k} mu(d) * mu(e) * 2^((n/d) * (k/e)).

Example

Array begins:
        2        2        6       12       30
        2       10       54      228      990
        6       54      498     4020    32730
       12      228     4020    65040  1047540
       30      990    32730  1047540 33554370

Mathematica

nn=5;
a[n_, k_]:=Sum[MoebiusMu[d]*MoebiusMu[e]*2^(n/d*k/e), {d, Divisors[n]}, {e, Divisors[k]}];
Table[a[n-k, k], {n, nn}, {k, n-1}]

Prog

(PARI) A(n, k) = {sumdiv(n, d, sumdiv(k, e, moebius(d) * moebius(e) * 2^((n/d) * (k/e))))} \\ Andrew Howroyd, Jan 19 2023

Crossrefs

First and last columns are A027375. Main diagonal is A265627.

Cf. A000031, A000740, A001037, A179043, A323351.

Cf. A323860, A323861, A323863, A323864, A323865, A323867, A323869.

Sequence in context: A121698 A087482 A137227 * A291667 A052537 A024945

Adjacent sequences: A323859 A323860 A323861 * A323863 A323864 A323865

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 04 2019

Status

approved