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%I #19 Aug 09 2018 10:23:22
%S 12,228,4020,65040,1047540,16768860,268419060,4294836480,68719210560,
%T 1099509531420,17592181850100,281474943095280,4503599560261620,
%U 72057593501073180,1152921503532053580,18446744065119682560,295147905162172956660,4722366482732189753280
%N Number of 3 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix.
%H Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, <a href="https://arxiv.org/abs/1602.06915">Periodicity in rectangular arrays</a>, arXiv:1602.06915 [cs.DM], 2016; Information Processing Letters 118 (2017) 58-63. See Table 1.
%t Psi[k_, m_, n_] := Sum[MoebiusMu[dm] MoebiusMu[dn] k^(m n/dm/dn), {dm, Divisors[m] }, {dn, Divisors[n]}];
%t Table[Psi[2, 4, n], {n, 1, 18}] (* _Jean-François Alcover_, Aug 09 2018, after _Lars Blomberg_ *)
%o (PARI) Psi(k,m,n) = v1=divisors(m); v2=divisors(n); sum(i1=1,length(v1),sum(i2=1,length(v2),moebius(v1[i1])*moebius(v2[i2])*k^(m*n/v1[i1]/v2[i2])));
%o vector(18,n,Psi(2,4,n)) \\ _Lars Blomberg_, Aug 19 2017
%Y Cf. A027375, A102536, A265627, A291070, A291071.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Aug 18 2017
%E a(8)-a(18) from _Lars Blomberg_, Aug 19 2017
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