OFFSET
1,3
COMMENTS
Three positive numbers a, b and c (without loss of generality c<=a, c<=b) define the faces of a disphenoid which can be used for a kaleidocycle if and only if -8*(a^2-b^2)^2*(a^2+b^2)-5*c^6+11*(a^2-b^2)^2*c^2+2*(a^2+b^2)*c^4>=0.
REFERENCES
Doris Schattschneider and Wallace Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0-906212-28-6
LINKS
H. M. Cundy and A. P. Rollett, Rotating rings of tetrahedra, in Mathematical Models, Oxford, 2nd. ed., 1961, pp. 144.
Herbert Kociemba, Kaleidocycles with 6 Disphenoids
Wikipedia, Kaleidocycle
EXAMPLE
a(3)=3 for example because there are three possible disphenoids with integer side lengths <=3: {2,2,1}, {3,3,1} and {3,3,2}. {1,1,1}, {2,2,2} and {3,3,3} define disphenoids (in this case regular tetrahedra) but the kaleidocycles will not work because the pieces block each other during the movement. {3,2,2} does not define a disphenoid because the faces of a disphenoid necessarily are acute triangles. And {2,1,1} , (3,1,1) and {3,2,1} do not even define triangles.
MATHEMATICA
value[a_, b_, c_]:=-8 (a^2-b^2)^2 (a^2+b^2)-5 c^6+11 (a^2-b^2)^2 c^2+2 (a^2+b^2) c^4
a[n_]:=Module[{a, b, c, t=0}, Do[If[value[a, b, c]>=0, t++], {c, n}, {b, c, n}, {a, b, n}]; t];
Array[a, 60]
CROSSREFS
KEYWORD
nonn
AUTHOR
Herbert Kociemba, Oct 22 2020
STATUS
approved