A138976 The discriminant of the characteristic polynomial of the O+ and O- submatrix for spin 3 of the nuclear electric quadrupole Hamiltonian is a perfect square for these values.

0, -3, -18, -45, -192, -459, -1914, -4557, -18960, -45123, -187698, -446685, -1858032, -4421739, -18392634, -43770717, -182068320, -433285443, -1802290578, -4289083725, -17840837472, -42457551819, -176606084154, -420286434477, -1748220004080, -4160406792963

Offset: 0

Author

Lorenz H. Menke, Jr., Mar 27 2008

Keywords

• sign
• easy

Comments

Perfect square values for discriminants are used to classify the Galois group of a polynomial. The O+ discriminant component is Sqrt[6(x^2-3x+6)] (used to generate these values) and for the O- discriminant Sqrt[6(x^2+3x+6)].

This sequence is the negative of the O+ sequence. Also, note that if 3*a[n] represents the positive terms, the negative terms are generated from 3 - 3*a[n].

For the O- sequence reverse the O+ sequence and change all of the signs to generate ...-446688, -45126, -18963, -4560, -1917, -462, -195, -48, -21, -6, -3, 0, 3, 18, 45, 192, 459, 1914, 4557, 18960, 45123, 187698, 446685.

Note that the difference equation a[n] generates the above sequence divided by 3 or ...,-148895, -62566, -15041, -6320, -1519, -638, -153, -64, -15, -6, -1, 0, 1, 2, 7, 16, 65, 154, 639, 15 20, 6321, 15042, 148896,...

This sequence, its reverse and the division by 3 form, all appear to be new.

References

• The physics reference is G. W. King, "The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels", Journal of Chemical Physics, Jan 1943, Volume 11, p27-42.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-1,1).

Crossrefs

A136331(n) = a(-n).

Programs

Mathematica

Do[If[IntegerQ[Sqrt[6 (6 - 3 x + x^2)]], Print[{x, Sqrt[6 (6 - 3 x + x^2)]}]], {x, -1000, 1000}]; Do[If[IntegerQ[Sqrt[6 (6 + 3 x + x^2)]], Print[{x, Sqrt[6 (6 + 3 x + x^2)]}]], {x, -1000, 1000}];

PARI

{a(n) = local(m); m = if( n>0, m = 1+n, -n); 3 * ((n>0) + (-1)^(n>0) * polcoeff( (x + x^2 - 5*x^3 - x^4) / ((1 - x) * (1 - 10*x^2 + x^4)) + x*O(x^m), m))} /* Michael Somos, Apr 05 2008 */

Formula

The difference equation is a[n]=11(a[n-2] - a[n-4])+a[n-6] with a[0]=0, a[1]=1, a[2]=2, a[3]=7, a[4]=16, a[5]=65. The solution is for even n: a[n]=(1/2) - (1/12)*(3+2*Sqrt[6])*(5-2*Sqrt[6])^(n/2)+(1/12)*(-3+2*Sqrt[6])*(5+2*Sqrt[6])^(n/2), for odd n a[n]=(1/2) - (1/12)*(3*Sqrt[2]+Sqrt[3])*(5-2*Sqrt[6])^(n/2)+(1/12)*(3*Sqrt[2]-Sqrt[3])*(5+2*Sqrt[6])^(n/2). Multiply the resultant sequence by 3 to generate the present sequence.

G.f.: -3*x*(1+5*x-x^2-x^3)/((1-x)*(1-10*x^2+x^4)). [Colin Barker, Aug 22 2012]