A265627 -id:A265627 - cp's OEIS frontend

A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

2, 2, 2, 6, 8, 6, 12, 54, 54, 12, 30, 216, 486, 216, 30, 54, 990, 4020, 4020, 990, 54, 126, 3912, 32730, 64800, 32730, 3912, 126, 240, 16254, 261414, 1047540, 1047540, 261414, 16254, 240, 504, 64800, 2097018, 16764840, 33554250, 16764840, 2097018, 64800, 504

Offset: 1

Gus Wiseman, Feb 04 2019

The 1-dimensional case is A027375.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Table begins:
       1     2     3     4
    ------------------------
  1: |  2     2     6    12
  2: |  2     8    54   216
  3: |  6    54   486  4020
  4: | 12   216  4020 64800
The A(2,2) = 8 arrays:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

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

First and last columns are A027375. Main diagonal is A323863.
Cf. A000740, A001037, A179043, A265627, A323351.
Cf. A323861, A323862, A323864, A323865, A323867, A323869.

# See A323861 for code.
for n in [1..8] do for k in [1..8] do Print(n*k*A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

T(n,k) = n*k*A323861(n,k). - Andrew Howroyd, Aug 21 2019

Terms a(29) and beyond from Andrew Howroyd, Aug 21 2019

A323869 Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 4, 24, 212, 1080, 18672, 94584, 2182752, 21261708, 408988080, 3245265144, 168549358368, 1053716696760, 42565371692592, 921132763909200, 26578273403903040, 260741534058271800, 20313207979498492344, 185603174638656822264, 16066126777465282744800, 324499299994016295338064

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(3) = 24 matrices:
  [123][132][213][312][231][321][122][211][112][221][121][212]
.
  [1][1][2][3][2][3][1][2][1][2][1][2]
  [2][3][1][1][3][2][2][1][1][2][2][1]
  [3][2][3][2][1][1][2][1][2][1][1][2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000670, A000740, A027375, A060223, A265627.

Cf. A323860, A323864, A323867, A323868, A323871.

List([1..30], A323869); # See A323861 for code; Andrew Howroyd, Aug 21 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[nrmmats[n],apermatQ]],{n,6}]

a(n) = n*A323871(n). - Andrew Howroyd, Aug 21 2019

Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019

A323864 Number of aperiodic binary arrays of size n.

Original entry on oeis.org

1, 2, 4, 12, 32, 60, 216, 252, 912, 1494, 3960, 4092, 23904, 16380, 65016, 130920, 324960, 262140, 1569132, 1048572, 6281280, 8388072, 16769016, 16777212, 134150880, 100663050, 268402680, 536865840, 1610449344, 1073741820, 8589664080, 4294967292, 25768888320

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(4) = 32 arrays:
  [0001][0010][0011][0100][0110][0111][1000][1001][1011][1100][1101][1110]
.
  [00] [00] [01] [01] [10] [10] [11] [11]
  [01] [10] [00] [11] [00] [11] [01] [10]
.
  [0] [0] [0] [0] [0] [0] [1] [1] [1] [1] [1] [1]
  [0] [0] [0] [1] [1] [1] [0] [0] [0] [1] [1] [1]
  [0] [1] [1] [0] [1] [1] [0] [0] [1] [0] [0] [1]
  [1] [0] [1] [0] [0] [1] [0] [1] [1] [0] [1] [0]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000740, A027375, A265627, A323351.

Cf. A323860, A323862, A323863, A323865, A323867, A323869.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[Length[Select[zaz[n],apermatQ]],{n,10}]

a(n) = Sum_{d|n} A323860(d, n/d). - Andrew Howroyd, Aug 21 2019

Terms a(18) and beyond from Andrew Howroyd, Aug 21 2019

A323863 Number of n X n aperiodic binary arrays.

Original entry on oeis.org

1, 2, 8, 486, 64800, 33554250, 68718675672, 562949953420302, 18446744060824780800, 2417851639229257812542976, 1267650600228226023797043513000, 2658455991569831745807614120560664598, 22300745198530623141521551172073990303938400

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(2) = 8 arrays are:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are (independently) aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000031, A000740, A027375, A179043, A265627, A323351.

Cf. A323860, A323862, A323864, A323865, A323867, A323869.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],apermatQ]],{n,4}]

a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - Robert Israel, Feb 04 2019

a(n) = n^2 * A323872(n). - Andrew Howroyd, Aug 21 2019

a(5) from Robert Israel, Feb 04 2019
a(6)-a(7) from Giovanni Resta, Feb 05 2019
Terms a(8) and beyond from Andrew Howroyd, Aug 21 2019

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.

Original entry on oeis.org

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

Gus Wiseman, Feb 04 2019

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

			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

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

First and last columns are A027375. Main diagonal is A265627.
Cf. A000031, A000740, A001037, A179043, A323351.
Cf. A323860, A323861, A323863, A323864, A323865, A323867, A323869.

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}]

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

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

A102536 Number of triangles similar to their n-th pedal, and not similar to any k-th pedal for k < n.

Original entry on oeis.org

2, 10, 54, 228, 990, 3966, 16254, 65040, 261576, 1046550, 4192254, 16768860, 67100670, 268402806, 1073708010, 4294836480, 17179738110, 68718948984, 274877382654, 1099509531420, 4398044397642, 17592177657846, 70368735789054, 281474943095280

Offset: 1

David W. Wilson, Jan 13 2005

nonn

The first pedal of a triangle has as its vertices the feet of the perpendiculars of the original triangle. The (n+1)st pedal is the pedal of the n-th pedal.
From Fortschritte JFM 34.0551.02 on the Valyi paper: The triangle with corners the altitude bases of a given triangle ABC are called pedal triangles. The pedal triangle of this triangle is the second pedal triangle. Generally, we understand the n-th pedal triangle of the triangle ABC to be the pedal triangle of the (n-1)th pedal triangle. The author searches for and counts all triangles that are similar to their n-th pedal triangle, where all mutually similar triangle are counted as one.
The number of these is psi(n)=2^n(2^n-1). The number of triangles for which the n-th pedal triangle is the first that is similar to it is Sum_{d|n} mu(n/d) psi(d), where mu is the Möbius function. The author ends with a table of those triangles that are similar to their first, 2nd and 3rd pedal triangles.
Also, the number of 2 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix; cf. A265627. - Jeffrey Shallit, Dec 11 2015

Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, Periodicity in rectangular arrays, Information Processing Letters 118 (2017) 58-63. See Table 1.
Hayashi, T. On the pedal triangles similar to the original triangles. Nieuw Archief (2) 10 (1912), 5-9. [Shows that there are 11 points whose pedal triangles are similar to the original triangle; those 11 points lie on a circle.]
de Vries, Jan, Über rechtwinklige Fusspunktdreiecke. Nieuw Archief (2) 9 (1910), 130-132. [The locus of those points that have rectangular pedal triangles with respect to a given triangle is determined by the three circles that cut the circumscribing circle orthogonally at two vertices of the triangle.]
Veldkamp, G. R. Classical geometry [Dutch], in Geometry, From Art to Science [Dutch], 1-15, CWI Syllabi, 33, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1993.

J. C. Alexander, The symbolic dynamics of the sequence of pedal triangles, Math. Mag. 66 (1993), no. 3, 147-158.
Jiu Ding, L. Richard Hitt, Xin-Min Zhang, Markov chains and dynamic geometry of polygons, Linear Algebra Appl. 367 (2003), 255-270.
John G. Kingston, John L. Synge, The sequence of pedal triangles, Amer. Math. Monthly 95 (1988), no. 7, 609-620.
J. H. Smith, Gyula Valyi [Source of sequence.]
Peter Ungar, Mixing property of the pedal mapping, Amer. Math. Monthly 97 (1990), no. 10, 898-900.
J. Valyi, Über die Fusspunktdreiecke, Monatsh. f. Math. 14 (1903), 243-252.

Cf. A027375, A265627, A290754, A291070.

Additional references supplied by Brendan McKay, Jan 14 2005
English summaries provided by Ralf Stephan, Jan 14 2005
More terms and formula from Valyi paper by Jeffrey Shallit, Nov 26 2015

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

N. J. A. Sloane, Aug 18 2017

nonn

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.

Cf. A027375, A102536, A265627, A291070, A291071.

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 *)

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

a(8)-a(18) from 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

N. J. A. Sloane, Aug 18 2017

nonn

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.

Cf. A027375, A102536, A265627, A290754, A291071.

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 *)

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

a(8)-a(16) from 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

N. J. A. Sloane, Aug 18 2017

nonn

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.

Cf. A027375, A102536, A265627, A290754, A291070.

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 *)

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

a(8)-a(15) from Lars Blomberg, Aug 19 2017

cp's OEIS Frontend

A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

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A323869 Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.

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A323864 Number of aperiodic binary arrays of size n.

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A323863 Number of n X n aperiodic binary arrays.

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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.

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A102536 Number of triangles similar to their n-th pedal, and not similar to any k-th pedal for k < n.

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A290754 Number of 3 X n binary matrices that are "primitive"; that is, they cannot be expressed as a "tiling" by a smaller matrix.

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