OFFSET
0,2
COMMENTS
The linked document "Proof Certificate" explains that period function T(v) measures precession of the J-vector along an algebraic sphere curve with local cyclic C_4 symmetry (also cf. Examples and A186375).
LINKS
W. G. Harter and C. W. Patterson, Rotational energy surfaces and high-J eigenvalue structure of polyatomic molecules, The Journal of Chemical Physics, 80 (1984), 4252.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Bradley Klee, Proof Certificate.
Bradley Klee, Checking Weierstrass data, 2023.
Eric Weisstein's World of Mathematics, Goursat's Surface.
FORMULA
3*n^2*a(n) - 4*(28*n^2-28*n+9)*a(n-1) + 64*(4*n-5)*(4*n-3)*a(n-2) = 0.
For n > 0, a(n) mod 3 = 0 (conjecture, tested up to n=3*10^6).
From Bradley Klee, May 30 2023: (Start)
The defining ODE can be derived from the following Weierstrass data:
g2 = (3/16)*(256 - 576*x + 405*x^2 - 81*x^3);
g3 = (1/64)*(4096 - 13824*x + 17496*x^2 - 9963*x^3 + 2187*x^4);
which determine an elliptic surface with four singular fibers. (End)
EXAMPLE
Period function T_{O}(w): Take T_{C3}(u) and T_{C4}(v) from A186375 and A318245 respectively. Set (u,v)=(w-2/3,2-w), with u in [0,1/3], v in [0,1], and w in [2/3,2]. Define piecewise function T_{O}(w) = T_{C3}(w-2/3) if w in [2/3,1] or T_{O}(w) = T_{C4}(2-w) if w in [1,2].
Geometric Singular Points: Construct a family of algebraic sphere curves by intersecting a sphere 1=X^2+Y^2+Z^2 with the octahedral surface w=2*(X^4+Y^4+Z^4). Four cube vertex axes--(x+y+z, -x+y+z, x-y+z, x+y-z)--intersect the sphere in eight circular points with w=2/3. Three octahedron vertex axes--(x, y, z)--intersect the sphere in six circular points with w=2. Six cuboctahedron vertex axes--(x+y, x-y, y+z, y-z, z+x, z-x)--intersect the sphere in twelve hyperbolic points with w=1.
MATHEMATICA
CoefficientList[Expand[Normal@Series[Divide[Sqrt[S], Sqrt[1-4*S*x]*Sqrt[S-8 + 8*Sqrt[1-4*S*x]]], {x, 0, 13}]/.{S->12+4*Q^2}]/.{Q^n_:>(1/2)^n*Binomial[n, n/2]} /.{x->1/3*x}, x]
RecurrenceTable[{3*n^2*a[n] - 4*(28*n^2-28*n+9)*a[n-1] + 64*(4*n-5)*(4*n-3)*a[n-2] == 0, a[0]==1, a[1]==12}, a, {n, 0, 1000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Bradley Klee, Aug 22 2018
STATUS
approved