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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.
1
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
0,2
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.
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]
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}];
PROG
(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 */
CROSSREFS
A136331(n) = a(-n).
Sequence in context: A337921 A365442 A094159 * A275038 A304976 A364599
KEYWORD
sign,easy
AUTHOR
STATUS
approved