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

%I A229461 #40 Jul 28 2025 20:17:45
%S A229461 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,
%T A229461 58,70,72,78,85,88,93,102,105,120,130,133,165,168,177,190,210,232,253,
%U A229461 273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848
%N A229461 Numbers k such that there is no convex pentagon that can be decomposed into k pairwise congruent regular equilateral triangles.
%C A229461 Conjecture: These 59 numbers are all such exceptions.
%C A229461 Terms are idoneal numbers (A000926) except for the six terms of A229462.
%C A229461 Numbers k not expressible as k = x^2 - y^2 - z^2 with x,y,z >= 1 and x > y+z.
%H A229461 Eike Hertel, <a href="http://www.minet.uni-jena.de/preprints/hertel_13/Regdreipfla.pdf">Reguläre Dreieckspflasterungen konvexer Polygone</a>, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/01/13, 2013 (preprint).
%H A229461 Eike Hertel and Christian Richter, <a href="https://doi.org/10.1007/s00454-014-9576-7">Tiling Convex Polygons with Congruent Equilateral Triangles</a>, Discrete Comput Geom (2014) 51:753-759.
%H A229461 E. Kani, <a href="http://www.labmath.uqam.ca/~annales/volumes/35-2/PDF/197-227.pdf">Idoneal numbers and some generalizations</a>, Ann Sci. Math. Québec, 35 (2011), pp. 197-227.
%H A229461 Kival Ngaokrajang, <a href="/A229461/a229461.pdf">Illustration of initial terms</a>
%Y A229461 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).
%K A229461 nonn
%O A229461 1,2
%A A229461 Suggested by Eike Hertel, _Hugo Pfoertner_, Sep 24 2013