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A136331 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, 6, 21, 48, 195, 462, 1917, 4560, 18963, 45126, 187701, 446688, 1858035, 4421742, 18392637, 43770720, 182068323, 433285446, 1802290581, 4289083728, 17840837475, 42457551822, 176606084157, 420286434480, 1748220004083 (list; graph; refs; listen; history; text; internal format)
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, 1520, 6321, 15042, 148896, ...

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

REFERENCES

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257.  Mathematical Reviews, MR2312537.  Zentralblatt MATH, Zbl 1133.11012.

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

Table of n, a(n) for n=0..25.

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 + x^2 - 5*x^3 - x^4) / (1 - x - 10*x^2 + 10*x^3 + x^4 - x^5). - Michael Somos, Apr 05 2008

a(n) = A138976(-n) for all n in Z. a(n) = 3 * A129444(n+1).

EXAMPLE

G.f. = 3*x + 6*x^2 + 21*x^3 + 48*x^4 + 195*x^5 + 462*x^6 + 1917*x^7 + ...

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) = my(m); m = if( n<0, m = 1-n, n); 3*(n<0) + 3*(-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

Cf. A129444, A138976.

Sequence in context: A076102 A094282 A124493 * A063683 A200380 A151396

Adjacent sequences:  A136328 A136329 A136330 * A136332 A136333 A136334

KEYWORD

nonn,uned

AUTHOR

Lorenz H. Menke, Jr., Mar 27 2008

EXTENSIONS

I rather feel that this should be broken up into two sequences, one each for the positive and negative terms, both starting at 0. - N. J. A. Sloane, Apr 04 2008

More terms from Michael Somos, Apr 05 2008

STATUS

approved

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Last modified October 20 08:05 EDT 2019. Contains 328252 sequences. (Running on oeis4.)