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%I #33 Jun 15 2023 03:29:47
%S 1,10,120,1540,20500,279480,3876600,54496200,774468900,11107261000,
%T 160553895040,2336799457200,34219387524400,503846306168800,
%U 7455357525594000,110811908027490960,1653792126235140900,24774309852363829800,372404448149589213600
%N 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).
%C 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).
%H É. Goursat, <a href="http://www.numdam.org/item/ASENS_1887_3_4__159_0">Étude des surfaces qui admettent tous les plans de symétrie d'un polyèdre régulier</a>, Annales scientifiques de l'École Normale Supérieure, Série 3 : Volume 4 (1887), 166-170.
%H W. G. Harter and D. E. Weeks, <a href="https://doi.org/10.1063/1.456659">Rotation-vibration spectra of icosahedral molecules. I. Icosahedral symmetry analysis and fine structure</a>, Journal of Chemical Physics, 90 (1989), 4370.
%H S. Herfurtner, <a href="https://doi.org/10.1007/BF01445211">Elliptic surfaces with four singular fibres</a>, Mathematische Annalen, 1991. <a href="https://archive.mpim-bonn.mpg.de/id/eprint/860/">Preprint</a>.
%H Bradley Klee, <a href="/A318495/a318495.pdf">Proof Certificate</a>.
%H Bradley Klee, <a href="/A318245/a318245_1.pdf">Checking Weierstrass data</a>, 2023.
%H O. Laporte, <a href="http://zfn.mpdl.mpg.de/search?keyword=Polyhedral_Harmonics_Laporte">Polyhedral Harmonics</a>, Zeitschrift für Naturforschung A, 8-11 (1948), 450.
%F 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.
%F For n > 0, a(n) mod 10 = 0 (conjecture, tested up to n=10^6).
%F From _Bradley Klee_, May 30 2023: (Start)
%F The defining ODE can be derived from the following Weierstrass data:
%F g2 = (243/256)*(256-640*x+520*x^2-135*x^3);
%F g3 = (729/8192)*(8192-30720*x+44160*x^2-29680*x^3+8775*x^4-729*x^5);
%F which determine an elliptic surface with four singular fibers. (End)
%e 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].
%e 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.
%t 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}]
%o (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
%Y Cf. A318496. Periods: A186375, A318245, A318417.
%K nonn
%O 0,2
%A _Bradley Klee_, Aug 27 2018
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