A338335 - OEIS

%I #21 Nov 16 2020 11:14:28

%S 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,

%T 151,173,196,220,246,273,304,337,373,411,452,496,541,589,640,692,748,

%U 807,871,939,1011,1086,1162,1242,1326,1413,1503,1597,1696,1799,1907,2018,2133,2254

%N 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.

%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 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 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.

%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+1,n},{a,b+1,n}];t];

%t Array[a,60]

%Y Cf. A338334, A338336.

%K nonn

%O 1,12

%A _Herbert Kociemba_, Oct 22 2020