A318495 Scaled g.f. T(u) = Sum_{n>=0} a(n)*(u/16)^n satisfies 5*(21*u-16)*T + d/du( 4*u*(u-1)*(27*u-32)*T') = 0, and a(0)=1; sequence gives a(n).

1, 10, 120, 1540, 20500, 279480, 3876600, 54496200, 774468900, 11107261000, 160553895040, 2336799457200, 34219387524400, 503846306168800, 7455357525594000, 110811908027490960, 1653792126235140900, 24774309852363829800, 372404448149589213600

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

Table of n, a(n) for n=0..18.

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

Cf. A318496. Periods: A186375, A318245, A318417.

Sequence in context: A024127 A005949 A027568 * A320760 A366711 A034255

Adjacent sequences: A318492 A318493 A318494 * A318496 A318497 A318498

Keyword

nonn

Author

Bradley Klee, Aug 27 2018

Status

approved