OFFSET
0,2
COMMENTS
The linked document "Proof Certificate" gives data for verifying that period function T(u) measures precession of the J-vector along an algebraic sphere curve with local cyclic C_5 symmetry (also cf. Examples and A318496).
LINKS
É. Goursat, Étude des surfaces qui admettent tous les plans de symétrie d'un polyèdre régulier, Annales scientifiques de l'École Normale Supérieure, Série 3 : Volume 4 (1887), 166-170.
W. G. Harter and D. E. Weeks, Rotation-vibration spectra of icosahedral molecules. I. Icosahedral symmetry analysis and fine structure, Journal of Chemical Physics, 90 (1989), 4370.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Bradley Klee, Proof Certificate.
Bradley Klee, Checking Weierstrass data, 2023.
O. Laporte, Polyhedral Harmonics, Zeitschrift für Naturforschung A, 8-11 (1948), 450.
FORMULA
2*n^2*a(n) - (59*n^2-59*n+20)*a(n-1) + 12*(6*n-7)*(6*n-5)*a(n-2) = 0.
For n > 0, a(n) mod 10 = 0 (conjecture, tested up to n=10^6).
From Bradley Klee, May 30 2023: (Start)
The defining ODE can be derived from the following Weierstrass data:
g2 = (243/256)*(256-640*x+520*x^2-135*x^3);
g3 = (729/8192)*(8192-30720*x+44160*x^2-29680*x^3+8775*x^4-729*x^5);
which determine an elliptic surface with four singular fibers. (End)
EXAMPLE
Period function T_{I}(w): Take T_{C5}(u) and T_{C3}(v) from A318495 and A318496 respectively. Set (u,v)=(1-w,w+5/27), with u in [0,1], v in [0,5/27], and w in [-5/27,1]. Define piecewise function T_{I}(w) = T_{C5}(1-w) if w in [0,1] or T_{I}(w) = T_{C3}(w+5/27) if w in [-5/27,0].
Geometric Singular Points: Construct a family of algebraic sphere curves by intersecting a sphere 1=X^2+Y^2+Z^2 with the icosahedral surface w=Z^6 - 5*(X^2+Y^2)*Z^4 + 5*(X^2+Y^2)^2*Z^2 - 2*(X^4-10*X^2*Y^2+5*Y^4)*X*Z. Six icosahedron vertex axes intersect the sphere in twelve circular points with w=1. Ten dodecahedron vertex axes intersect the sphere in twenty circular points with w=-5/27. Fifteen icosidodecahedron vertex axes intersect the sphere in thirty hyperbolic points with w=0.
MATHEMATICA
RecurrenceTable[{2 n^2 a[n] - (59 n^2 - 59 n + 20) a[n - 1] + 12 (6 n - 7) (6 n - 5) a[n - 2] == 0, a[0] == 1, a[1] == 10}, a, {n, 0, 1000}]
PROG
(GAP) a:=[1, 10];; for n in [3..20] do a[n]:=(1/(2*(n-1)^2))*(( (59*(n^2-3*n+2)+20)*a[n-1]-(12*(6*n-13)*(6*n-11))*a[n-2])); od; a; # Muniru A Asiru, Sep 24 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Bradley Klee, Aug 27 2018
STATUS
approved