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A136331
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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.
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1
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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
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OFFSET
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0,2
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COMMENTS
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Perfect square values for discriminants are used to classify the Galois group of a polynomial. The O+ discriminant component is sqrt(6*(x^2-3*x+6)) (used to generate these values) and for the O- discriminant sqrt(6*(x^2+3*x+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.
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REFERENCES
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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, pp. 27-42.
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LINKS
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FORMULA
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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 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 even 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) for odd n. 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
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EXAMPLE
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G.f. = 3*x + 6*x^2 + 21*x^3 + 48*x^4 + 195*x^5 + 462*x^6 + 1917*x^7 + ...
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MATHEMATICA
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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}];
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,uned
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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