A290754
Number of 3 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix.
Original entry on oeis.org
12, 228, 4020, 65040, 1047540, 16768860, 268419060, 4294836480, 68719210560, 1099509531420, 17592181850100, 281474943095280, 4503599560261620, 72057593501073180, 1152921503532053580, 18446744065119682560, 295147905162172956660, 4722366482732189753280
Offset: 1
- Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, Periodicity in rectangular arrays, arXiv:1602.06915 [cs.DM], 2016; Information Processing Letters 118 (2017) 58-63. See Table 1.
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Psi[k_, m_, n_] := Sum[MoebiusMu[dm] MoebiusMu[dn] k^(m n/dm/dn), {dm, Divisors[m] }, {dn, Divisors[n]}];
Table[Psi[2, 4, n], {n, 1, 18}] (* Jean-François Alcover, Aug 09 2018, after Lars Blomberg *)
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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])));
vector(18,n,Psi(2,4,n)) \\ Lars Blomberg, Aug 19 2017
A291070
Number of 4 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix.
Original entry on oeis.org
30, 990, 32730, 1047540, 33554370, 1073708010, 34359738210, 1099510578960, 35184372055560, 1125899873286210, 36028797018961890, 1152921503532053580, 36893488147419095010, 1180591620683051547810, 37778931862957128089670, 1208925819613529663013120
Offset: 1
- Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, Periodicity in rectangular arrays, arXiv:1602.06915 [cs.DM], 2016; Information Processing Letters 118 (2017) 58-63. See Table 1.
-
Psi[k_, m_, n_] := Sum[MoebiusMu[dm] MoebiusMu[dn] k^(m n/dm/dn), {dm, Divisors[m] }, {dn, Divisors[n]}];
Table[Psi[2, 5, n], {n, 1, 16}] (* Jean-François Alcover, Aug 09 2018, after Lars Blomberg *)
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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])));
vector(16,n,Psi(2,5,n)) \\ Lars Blomberg, Aug 19 2017
A291071
Number of 5 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix.
Original entry on oeis.org
54, 3966, 261522, 16768860, 1073708010, 68718945018, 4398044397642, 281474943095280, 18014398374741048, 1152921502458345570, 73786976286244079562, 4722366482732172984420, 302231454903107470761930, 19342813113825270435966978, 1237940039285345088379356750
Offset: 1
- Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, Periodicity in rectangular arrays, arXiv:1602.06915 [cs.DM], 2016; Information Processing Letters 118 (2017) 58-63. See Table 1.
-
Psi[k_, m_, n_] := Sum[MoebiusMu[dm] MoebiusMu[dn] k^(m n/dm/dn), {dm, Divisors[m] }, {dn, Divisors[n]}];
Table[Psi[2, 6, n], {n, 1, 15}] (* Jean-François Alcover, Aug 10 2018, after Lars Blomberg *)
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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])));
vector(15,n,Psi(2,6,n)) \\ Lars Blomberg, Aug 19 2017
Showing 1-3 of 3 results.