A229461 Numbers k such that there is no convex pentagon that can be decomposed into k pairwise congruent regular equilateral triangles.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 16, 18, 21, 22, 24, 25, 30, 33, 37, 40, 42, 45, 48, 57, 58, 70, 72, 78, 85, 88, 93, 102, 105, 120, 130, 133, 165, 168, 177, 190, 210, 232, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

Offset: 1

Author

Suggested by Eike Hertel, Hugo Pfoertner, Sep 24 2013

Keywords

• nonn

Comments

Conjecture: These 59 numbers are all such exceptions.

Terms are idoneal numbers (A000926) except for the six terms of A229462.

Numbers k not expressible as k = x^2 - y^2 - z^2 with x,y,z >= 1 and x > y+z.

Links

• Eike Hertel, Reguläre Dreieckspflasterungen konvexer Polygone, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/01/13, 2013 (preprint).
• Eike Hertel and Christian Richter, Tiling Convex Polygons with Congruent Equilateral Triangles, Discrete Comput Geom (2014) 51:753-759.
• E. Kani, Idoneal numbers and some generalizations, Ann Sci. Math. Québec, 35 (2011), pp. 197-227.
• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A000926 (idoneal numbers), A229462 (idoneal numbers not in this sequence), A229757 (hexagon exception numbers), A025052 (numbers not of form a*b+b*c+c*a).