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

Author

Bradley Klee, Aug 27 2018

Keywords

• nonn

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).

Examples

			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.
		

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.

Crossrefs

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

Programs

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

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}]

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)

G.f.: hypergeom([1/12, 5/12],[1],1728*x^5*(27*x-2)^3*(16*x-1)^2/(2160*x^3-520*x^2+40*x-1)^3)/(1-40*x+520*x^2-2160*x^3)^(1/4). - Mark van Hoeij, Dec 13 2024