OFFSET
1,12
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
Herbert Kociemba, Kaleidocycles with 6 Disphenoids
Wikipedia, Kaleidocycle
EXAMPLE
For example a(11)=1 and a(n)=0 for n<11 because the only scalene possible triangle face for a disphenoid with edge lengths <=11 is {11,10,8}. All other possible triples define disphenoids where the kaleidocycle movement blocks at some point or do not define disphenoids at all.
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+1, n}, {a, b+1, n}]; t];
Array[a, 60]
CROSSREFS
KEYWORD
nonn
AUTHOR
Herbert Kociemba, Oct 22 2020
STATUS
approved