A338335 Number of integer-sided disphenoids (isosceles tetrahedra) with scalene triangle faces and side lengths <=n which can be used to build a kaleidocycle by connecting 6 congruent pieces into a cycle.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 10, 14, 19, 25, 32, 40, 48, 57, 68, 82, 97, 113, 131, 151, 173, 196, 220, 246, 273, 304, 337, 373, 411, 452, 496, 541, 589, 640, 692, 748, 807, 871, 939, 1011, 1086, 1162, 1242, 1326, 1413, 1503, 1597, 1696, 1799, 1907, 2018, 2133, 2254

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

Table of n, a(n) for n=1..60.

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

Cf. A338334, A338336.

Sequence in context: A282731 A134919 A033437 * A226185 A310071 A330259

Adjacent sequences: A338332 A338333 A338334 * A338336 A338337 A338338

Keyword

nonn

Author

Herbert Kociemba, Oct 22 2020

Status

approved