A102536 - OEIS

%I #32 Aug 18 2017 11:32:10

%S 2,10,54,228,990,3966,16254,65040,261576,1046550,4192254,16768860,

%T 67100670,268402806,1073708010,4294836480,17179738110,68718948984,

%U 274877382654,1099509531420,4398044397642,17592177657846,70368735789054,281474943095280

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

%C 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.

%C 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.

%C 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.

%C 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

%D Guilhem Gamard, Gwenaël Richomme, Jeffrey Shallit, Taylor J. Smith, Periodicity in rectangular arrays, Information Processing Letters 118 (2017) 58-63. See Table 1.

%D 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.]

%D 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.]

%D 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.

%H J. C. Alexander, <a href="http://www.jstor.org/stable/2690959">The symbolic dynamics of the sequence of pedal triangles</a>, Math. Mag. 66 (1993), no. 3, 147-158.

%H Jiu Ding, L. Richard Hitt, Xin-Min Zhang, <a href="http://dx.doi.org/10.1016/S0024-3795(02)00634-1">Markov chains and dynamic geometry of polygons</a>, Linear Algebra Appl. 367 (2003), 255-270.

%H John G. Kingston, John L. Synge, <a href="http://www.jstor.org/stable/2323303">The sequence of pedal triangles</a>, Amer. Math. Monthly 95 (1988), no. 7, 609-620.

%H J. H. Smith, <a href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Valyi.html">Gyula Valyi</a> [Source of sequence.]

%H Peter Ungar, <a href="http://www.jstor.org/stable/2324326">Mixing property of the pedal mapping</a>, Amer. Math. Monthly 97 (1990), no. 10, 898-900.

%H J. Valyi, <a href="http://dx.doi.org/10.1007/BF01706871">Über die Fusspunktdreiecke</a>, Monatsh. f. Math. 14 (1903), 243-252.

%Y Cf. A027375, A265627, A290754, A291070.

%K nonn

%O 1,1

%A _David W. Wilson_, Jan 13 2005

%E Additional references supplied by _Brendan McKay_, Jan 14 2005

%E English summaries provided by _Ralf Stephan_, Jan 14 2005

%E More terms and formula from Valyi paper by _Jeffrey Shallit_, Nov 26 2015