A338336 Number of integer-sided disphenoids (isosceles tetrahedra) with triangle face perimeter n which can be used to build a kaleidocycle by connecting 6 congruent pieces into a cycle.

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 6, 5, 6, 6, 7, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 9, 11, 10, 11, 11, 12, 11, 12, 12, 14, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22

Offset

1,11

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..80.

Herbert Kociemba, Kaleidocycles with 6 Disphenoids

Wikipedia, Kaleidocycle

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[a=n-b-c; a>=b&&value[a, b, c]>=0, t++], {c, Quotient[n, 3]}, {b, c, n-c}]; t];
Array[a, 80]

Crossrefs

Cf. A338334, A338335.

Sequence in context: A261348 A320536 A347698 * A298783 A053280 A289122

Adjacent sequences: A338333 A338334 A338335 * A338337 A338338 A338339

Keyword

nonn

Author

Herbert Kociemba, Oct 22 2020

Status

approved