cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

David W. Wilson, Jan 13 2005

Keywords

Comments

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

References

  • 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.

Links

Crossrefs

Cf. A027375, A265627, A290754, A291070.

Extensions

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

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

Views

Author

N. J. A. Sloane, Aug 18 2017

Keywords

Links

  • 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.

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • 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])));
vector(16,n,Psi(2,5,n)) \\ Lars Blomberg, Aug 19 2017

Extensions

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

Views

Author

N. J. A. Sloane, Aug 18 2017

Keywords

Links

  • 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.

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • 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])));
vector(15,n,Psi(2,6,n)) \\ Lars Blomberg, Aug 19 2017

Extensions

a(8)-a(15) from Lars Blomberg, Aug 19 2017
Showing 1-3 of 3 results.

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