A338334 Number of integer-sided disphenoids (isosceles tetrahedra) with side lengths <= n which can be used to build a kaleidocycle by connecting 6 congruent pieces into a cycle.

0, 1, 3, 6, 10, 15, 21, 28, 36, 44, 54, 66, 80, 96, 113, 132, 153, 176, 200, 225, 252, 282, 316, 352, 390, 431, 475, 522, 570, 620, 673, 728, 788, 851, 918, 988, 1062, 1139, 1218, 1301, 1388, 1477, 1571, 1669, 1773, 1882, 1996, 2113, 2232, 2356, 2485, 2618, 2755, 2897, 3045, 3198, 3356, 3518, 3685, 3859

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

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

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

Cf. A338335, A338336.

Sequence in context: A363777 A108923 A033441 * A107082 A267238 A256379

Adjacent sequences: A338331 A338332 A338333 * A338335 A338336 A338337

Keyword

nonn

Author

Herbert Kociemba, Oct 22 2020

Status

approved