A338334 - OEIS

%I #17 Nov 16 2020 11:14:56

%S 0,1,3,6,10,15,21,28,36,44,54,66,80,96,113,132,153,176,200,225,252,

%T 282,316,352,390,431,475,522,570,620,673,728,788,851,918,988,1062,

%U 1139,1218,1301,1388,1477,1571,1669,1773,1882,1996,2113,2232,2356,2485,2618,2755,2897,3045,3198,3356,3518,3685,3859

%N 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.

%C 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.

%D Doris Schattschneider and Wallace Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0-906212-28-6

%H H. M. Cundy and A. P. Rollett, <a href="https://archive.org/details/MathematicalModels-/page/n147/mode/2up">Rotating rings of tetrahedra</a>, in Mathematical Models, Oxford, 2nd. ed., 1961, pp. 144.

%H Herbert Kociemba, <a href="http://kociemba.org/themen/kaleidocycles/intro.html">Kaleidocycles with 6 Disphenoids</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kaleidocycle">Kaleidocycle</a>

%e 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.

%t 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

%t 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];

%t Array[a,60]

%Y Cf. A338335, A338336.

%K nonn

%O 1,3

%A _Herbert Kociemba_, Oct 22 2020