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A102536
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Number of triangles similar to their n-th pedal, and not similar to any k-th pedal for k < n.
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3
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2, 10, 54, 228, 990, 3966, 16254, 65040, 261576, 1046550, 4192254, 16768860, 67100670, 268402806, 1073708010, 4294836480, 17179738110, 68718948984, 274877382654, 1099509531420, 4398044397642, 17592177657846, 70368735789054, 281474943095280
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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